Abstract:

An optical communications link is described, comprising first and second
fiber lines in substantial scaled translational symmetry by a common
scaling factor with respect to a second-order dispersion coefficient
profile (oppositely signed) and with respect to at least one of a
loss/gain coefficient profile and a nonlinear coefficient-power product
profile for facilitating progressive compensation along the second fiber
line of at least one nonlinearity introduced along the first fiber line.

Claims:

1. An optical communications link, comprising: a contiguous series
arrangement of N-1 fiber spans, 2.ltoreq.N-1<10, each fiber span i=1,
. . . , N-1 comprising a primary fiber line "i" characterized by an
ith parameter set [αi(z), β2,i(z),
(γgP)i(z)] in which αi(z) is a loss/gain
coefficient profile, β2,i(z) is a second-order dispersion
coefficient profile, and (γgP)i(z) is a first nonlinear
coefficient-power profile; and an Nth fiber span comprising a
primary fiber line "N" characterized by an Nth parameter set
[αN(z), β2,N(z), (γgP)N(z)]; wherein
along each primary fiber line "i" there is a relationship
Ri[αi(Riz), -.beta.2,i(Riz),
(γgP)i(Riz)]≈[αN(z),
β2,N(z), (γgP)N(z)] with Ri being a real
scalar constant, for facilitating compensation of at least one nonlinear
effect in an optical signal propagating through said N fiber spans.

3. The optical communications link of claim 1, further comprising an
optical phase conjugator positioned between said (N-1)th fiber span
and said Nth fiber span.

4. The optical communications link of claim 1, wherein each of said N
fiber spans further comprises a secondary fiber line "i" connected to
said primary fiber line "i" and characterized by respective continuations
of said ith parameter set [αi(z), β2,i(z),
(γgP)i(z)] thereof, each secondary fiber line "i" being
connected to said primary fiber line "i" at an ith location where
(γgP)i(z) becomes generally small compared to a maximum
value [(γgP)i(z)]MAX thereof, wherein along each
secondary fiber line "i" said relationship Ri[ai(Riz),
-.beta.2,i(Riz),
(γgP)i(Riz)]≈[αN(z),
β2,N(z), (γgP)N(z)] is not satisfied, said
non-satisfaction accommodating length variations in said N secondary
fiber lines designed to achieve predetermined target values for
accumulated second-order dispersion for each of said N fiber spans, said
non-satisfaction not substantially confounding results of said
compensation of said at least one nonlinear effect due to relatively low
power levels in said N secondary fiber lines.

5. The optical communications link of claim 4, wherein each of said N
primary fiber lines is a transmission single-mode fiber having a constant
loss/gain coefficient profile value less than 0.5 therealong, and wherein
each of said N secondary fiber lines is a dispersion compensating fiber
having a constant loss/gain coefficient profile value greater than 1.0.

6. The optical communications link of claim 1, wherein Ri is between
0.05 and 20 for each of said N-1 fiber spans.

7. An optical communications link for guiding a wavelength-division
multiplexed (WDM) optical signal between a first node and a second node,
the optical signal having a plurality of channels, comprising: a first
fiber span receiving the optical signal from the first node and having
Kerr nonlinear effects introducing ghost-pulse artifacts into at least
one of the channels, said first fiber span being dispersion-compensated;
a second fiber span transferring the optical signal to the second node
and having Kerr nonlinear effects similar to those of said first fiber
span introducing similar ghost-pulse artifacts into said at least one
channel; and a self-phase modulating device positioned between said first
and second fiber spans, comprising an array of self-phase modulators
corresponding respectively to each of said plurality of channels
configured such that said ghost-pulse artifacts introduced by said first
fiber span are substantially reduced upon arrival of said optical signal
at said second node.

8. The optical communications link of claim 7, each of said self-phase
modulators introducing an approximately 180-degree phase shift at a
nominal power level of pulse peaks of an associated one of said plurality
of channels.

9. The optical communications link of claim 7, said self-phase modulating
device comprising a WDM demultiplexer coupling said first fiber span to
said self-phase modulators and a WDM multiplexer coupling said self-phase
modulators to said second fiber span.

11. The optical communications link of claim 10, wherein said self-phase
modulators each have a dispersion characteristic designed to limit
spectral broadening of the phase-modulated pulses in the associated
channel.

12. An integrated dispersion-compensating module for installation at an
electrically powered amplifying location of an optical communications
link, the optical communications link having first and second
transmission fiber lines of known first and second lengths, known first
and second loss/gain coefficient profiles, and known first and second
second-order dispersion coefficient profiles, respectively, the
integrated dispersion-compensating module comprising: a first
dispersion-compensating fiber line connected to said first transmission
fiber line and having a third loss/gain coefficient profile designed for
substantial scaled translational symmetry with said second loss/gain
coefficient profile and a third second-order dispersion coefficient
profile designed for substantial scaled translational symmetry with said
second second-order dispersion coefficient profile oppositely signed by
said first constant; and a second dispersion-compensating fiber line
connected to said first dispersion-compensating fiber line; wherein said
second dispersion-compensating fiber line has a length selected such that
an accumulated dispersion associated with said first and second
dispersion-compensating fiber lines compensates an accumulated dispersion
associated with said first transmission fiber line within a first
predetermined tolerance for said known length of said first transmission
fiber line.

13. The integrated dispersion-compensating module of claim 12, said
second transmission fiber line being further characterized by a first
nonlinear coefficient profile at a first nominal input power level,
wherein said first dispersion-compensating fiber line has a second
nonlinear coefficient profile designed for substantial scaled
translational symmetry with said first nonlinear coefficient profile by a
second constant.

14. The integrated dispersion-compensating module of claim 12, further
comprising third and fourth dispersion-compensating fiber lines commonly
packaged with said first and second dispersion-compensating fiber lines,
said third dispersion-compensating fiber line for connection to said
second transmission fiber line and designed to have a fourth loss/gain
coefficient profile in substantial scaled translational symmetry with
said first loss/gain coefficient profile by a third constant and a fourth
second-order dispersion profile in substantial scaled translational
symmetry with said first second-order dispersion coefficient profile
oppositely signed by said fourth constant, said fourth
dispersion-compensating fiber line being connected to said third
dispersion-compensating fiber line, wherein said fourth
dispersion-compensating fiber line has a length selected such that an
accumulated dispersion associated with said third and fourth
dispersion-compensating fiber lines compensates an accumulated dispersion
associated with said second transmission fiber line within a second
predetermined tolerance for said known length of said second transmission
fiber line.

15. The integrated dispersion-compensating module of claim 12, said
optical communications link being designed to propagate an optical signal
in a direction of propagation from said first transmission fiber line
toward said second transmission fiber line, wherein said integrated
dispersion-compensating module is configured for installation prior to
said first transmission fiber line relative to said direction of
propagation.

16. The integrated dispersion-compensating module of claim 12, said
optical communications link being designed to propagate an optical signal
in a direction from said first transmission fiber line toward said second
transmission fiber line, wherein said integrated dispersion-compensating
module is configured for installation between said first and second
transmission fiber lines relative to said direction of propagation.

Description:

CROSS-REFERENCE TO RELATED APPLICATIONS

[0001] This application is a continuation of U.S. Ser. No. 11/173,505,
filed on Jul. 1, 2005 and currently pending as of the filing date of this
application, and which claims the benefit of Provisional Application Ser.
No. 60/585,270, filed Jul. 2, 2004. Each of the above-referenced
applications is incorporated by reference herein.

[0004] Optical fiber nonlinearities begin to manifest themselves as the
capabilities of the channel are pushed to their limits through the use of
increased signal power, higher bit rates, longer transmission distances,
and increased numbers of channels. One physical mechanism associated with
at least one fiber nonlinearity is the optical Kerr effect, in which the
refractive index of an optical fiber varies in accordance with the
intensity of an optical signal. The variation of the refractive index
modulates the phase of the optical signal, resulting in adverse effects
such as self-phase modulation (SPM), cross-phase modulation (XPM), and
four-wave mixing (FWM). Another physical mechanism associated with at
least one fiber nonlinearity is the Raman effect, arising from energy
transfers between the propagating photons and the vibrational/rotational
modes of the glass molecules in the fiber.

[0005] Because of fiber nonlinearities, there may be substantial
restrictions on one or more of signal power, the number of WDM channels
that can be carried, bit rates per channel, permissible fiber dispersion
amounts, and maximum regenerative repeater spacings. It would be
desirable to provide an optical fiber communications system in which
nonlinearities induced by optical fibers are at least partially
compensated, while also providing for the necessary dispersion
compensation. It would be further desirable to provide such optical fiber
communications system using fiber spans that can be physically realized
using known, off-the-shelf optical components. Other issues arise as
would be apparent to apparent to one skilled in the art upon reading the
present disclosure.

SUMMARY

[0006] An optical communications link is provided, comprising first and
second fiber lines in substantial scaled translational symmetry by a
common scaling factor with respect to a second-order dispersion
coefficient profile (oppositely signed) and with respect to at least one
of a loss/gain coefficient profile and a nonlinear coefficient-power
product profile for facilitating progressive compensation along the
second fiber line of at least one nonlinearity introduced along the first
fiber line. In one embodiment, the substantial scaled translational
symmetry by the common scaling factor is characterized in that, for a
first profile and a second profile, the first profile is in substantial
scaled translational symmetry by the common scaling factor with the
second profile if the first profile, when expanded along a first axis by
the common scaling factor and contracted along a second axis by the
common scaling factor, is in substantial correspondence with the second
profile.

[0007] Also provided is an optical communications link, comprising a first
fiber span including a first fiber line, the first fiber line comprising
a first fiber segment, and a second fiber span including a second fiber
line, the second fiber line comprising a second fiber segment. The first
and second fiber lines are in substantial scaled translational symmetry
by a first common scaling factor along the first and second fiber
segments with respect to a second-order dispersion coefficient profile
(oppositely signed) and with respect to at least one of a loss/gain
coefficient profile and a nonlinear coefficient-power product profile for
facilitating progressive compensation along the second fiber segment of
at least one nonlinearity introduced along the first fiber segment.

[0008] Also provided is an optical communications link, comprising first
and second fiber lines characterized by a loss/gain coefficient profile
pair, a second-order dispersion coefficient profile pair, and a nonlinear
coefficient-power product profile pair. For facilitating progressive
compensation along the second fiber line of at least one nonlinearity
introduced along the first fiber line, the first and second fiber lines
are configured such that, for the second-order dispersion coefficient
profile pair and at least one of the loss/gain coefficient profile pair
and the nonlinear coefficient-power product profile pair, a first profile
thereof substantially corresponds to a second profile thereof when the
first profile is expanded along a first axis by a common scaling factor
and contracted along a second axis by that common scaling factor.

[0009] Also provided is an optical communications link, comprising a first
fiber span including a first fiber line, the first fiber line comprising
a first fiber segment and having a first loss/gain coefficient profile, a
first second-order dispersion coefficient profile, and a first nonlinear
coefficient-power product profile. The optical communications link
further comprises a second fiber span including a second fiber line, the
second fiber line comprising a second fiber segment and having a second
loss/gain coefficient profile, a second second-order dispersion
coefficient profile, and a second nonlinear coefficient-power product
profile. The optical communications link further comprises an optical
phase conjugator optically coupled between the first and second fiber
spans, an optical signal received by the first fiber span being
propagated respectively through the first fiber span, the optical phase
conjugator, and the second fiber span. The first fiber span, the second
fiber span, and the optical phase conjugator are configured such that
each of the following three conditions is satisfied: (i) the second
loss/gain coefficient profile along the second fiber segment is in
substantial scaled translational symmetry with the first loss/gain
coefficient profile along the first fiber segment by a first constant;
(ii) the second second-order dispersion coefficient profile along the
second fiber segment is in substantial scaled translational symmetry with
the first second-order dispersion coefficient profile oppositely signed
along the first fiber segment by the first constant, and (iii) the second
nonlinear coefficient-power product profile along the first fiber segment
is in substantial scaled translational symmetry with the first nonlinear
coefficient-power product profile along the second fiber segment by the
first constant. Progressive compensation along the second fiber segment
of at least one nonlinearity introduced along the first fiber segment is
facilitated. In other embodiments, the optical phase conjugator may be
omitted.

[0010] Also provided is an optical communications link, comprising a
contiguous series arrangement of N-1 fiber spans, 2≦N-1<10,
each fiber span i=1, N-1 comprising a primary fiber line "i"
characterized by an ith parameter set [αi(z),
β2,i(z), (γgP)i(z)] in which αi(z)
is a loss/gain coefficient profile, β2,i(z) is a second-order
dispersion coefficient profile, and (γgP)i(z) is a first
nonlinear coefficient-power profile. The optical communications link
further comprises an Nth fiber span comprising a primary fiber line
"N" characterized by an Nth parameter set [αN(z),
β2,N(z), (γgP)N(z)]. Along each primary fiber
line "i" there is a relationship Ri[αi(Riz),
-β2,i(Riz),
(γgP)i(Riz)]≈[αN(z),
β2,N(z), (γgP)N(z)] with Ri being a real
scalar constant, for facilitating compensation of at least one nonlinear
effect in an optical signal propagating through the N fiber spans.

[0011] Also provided is an optical communications link for guiding a
wavelength-division multiplexed (WDM) optical signal between a first node
and a second node, the optical signal having a plurality of channels. The
optical communications link comprises a first fiber span receiving the
optical signal from the first node and having Kerr nonlinear effects
introducing ghost-pulse artifacts into at least one of the channels. The
first fiber span is dispersion-compensated. The optical communications
link further comprises a second fiber span transferring the optical
signal to the second node and having Kerr nonlinear effects similar to
those of the first fiber span introducing similar ghost-pulse artifacts
into the at least one channel. A self-phase modulating device is
positioned between the first and second fiber spans, comprising an array
of self-phase modulators corresponding respectively to each of the
plurality of channels configured such that the ghost-pulse artifacts
introduced by the first fiber span are substantially reduced upon arrival
of the optical signal at the second node.

[0012] Also provided is an optical fiber pair for use in a
nonlinearity-managed optical communications link, comprising a first
optical fiber and a second optical fiber. The first optical fiber is an
off-the-shelf optical fiber designed for long-distance transport of
optical signals with minimized attenuation. The first optical fiber is
characterized by a first loss coefficient and a first second-order
dispersion coefficient. The second optical fiber is designed for
dispersion compensation and has a second loss coefficient and a second
second-order dispersion coefficient. The second optical fiber is
fabricated such that a first ratio of the second second-order dispersion
coefficient to the second loss coefficient is substantially equal to a
second ratio of the first second-order dispersion coefficient to the
first loss coefficient oppositely signed.

[0013] Also provided is an integrated dispersion-compensating module for
installation at an electrically powered amplifying location of an optical
communications link having first and second transmission fiber lines of
known first and second lengths, known first and second loss/gain
coefficient profiles, and known first and second second-order dispersion
coefficient profiles, respectively. The integrated
dispersion-compensating module comprises a first dispersion-compensating
fiber line that is connected to the first transmission fiber line and
that has a third loss/gain coefficient profile designed for substantial
scaled translational symmetry with the second loss/gain coefficient
profile. The first dispersion-compensating fiber line also has a third
second-order dispersion coefficient profile designed for substantial
scaled translational symmetry with the second second-order dispersion
coefficient profile, oppositely signed, by the first constant. The
integrated dispersion-compensating module further comprises a second
dispersion-compensating fiber line connected to the first
dispersion-compensating fiber line. The second dispersion-compensating
fiber line has a length selected such that an accumulated dispersion
associated with the first and second dispersion-compensating fiber lines
compensates an accumulated dispersion associated with the first
transmission fiber line within a first predetermined tolerance for the
known length of the first transmission fiber line.

BRIEF DESCRIPTION OF THE DRAWINGS

[0014]FIG. 1 illustrates an optical communications system according to an
embodiment;

[0015] FIG. 2 illustrates an optical communications link according to an
embodiment;

[0016]FIG. 3 illustrates an optical communications link and associated
profiles according to an embodiment;

[0017] FIGS. 4-9 illustrate optical communications links according to one
or more embodiments;

[0018] FIG. 10 illustrates two fiber spans in translational symmetry about
an optical phase conjugator, the shaded areas represent two typical fiber
segments that are in scaled translational symmetry about the conjugator;

[0019] FIG. 11 illustrates the cascade of two fiber lines with opposite
nonlinear coefficients but identical linear parameters of dispersion and
loss/gain, DCG=dispersion compensation and gain;

[0020] FIG. 12 illustrates the signal power and dispersion maps for a
series of two fiber lines with opposite nonlinear coefficients but
identical linear parameters of dispersion and loss/gain;

[0021]FIG. 13 illustrates that the functionality of a fictitious fiber
with negative nonlinearities may be realized equivalently by a
conventional fiber with positive nonlinearities with the help of OPC;

[0022] FIG. 14 illustrates a mirror-symmetric configuration of pairs of
fiber spans in scaled translational symmetry, with the dispersion in each
span compensated to zero. Top: schematic arrangement of fibers and
amplifiers with respect to OPC. Middle: map of signal power P(z) along
the propagation distance z. Bottom: map of accumulated dispersion b2(z)
along the propagation distance z;

[0023] FIG. 15 illustrates a mirror-symmetric configuration of pairs of
fiber spans in scaled translational symmetry, with non-zero residual
dispersion in the spans. There are pre- and post-dispersion compensators
(DCs), as well as a dispersion conditioner immediately after OPC. Top:
schematic arrangement of fibers and amplifiers with respect to OPC.
Middle: map of signal power P(z) along the propagation distance z.
Bottom: map of accumulated dispersion b2(z) along the propagation
distance z;

[0024] FIG. 16 illustrates a mirror-symmetric configuration of pairs of
fiber spans in scaled translational symmetry, with non-zero residual
dispersion in the spans. There are pre- and post-dispersion compensators
(DCs) but no dispersion conditioner at the site of OPC. Top: schematic
arrangement of fibers and amplifiers with respect to OPC. Middle: map of
signal power P(z) along the propagation distance z. Bottom: map of
accumulated dispersion b2(z) along the propagation distance z;

[0025]FIG. 17 illustrates a transmission line consisting of SMFs and
slope-matching DCFs;

[0026] FIG. 18 illustrates received eye diagrams of the 2nd DEMUX channel.
Top row: transmission results of the setup in FIG. 17. Top-left: fiber
nonlinearity is OFF, the signal is only impaired by amplifier noise.
Top-right: fiber nonlinearity is ON, the signal distortion is only
increased slightly. Bottom row: transmission results when the setup is
modified, and the fiber nonlinearity is always ON. Bottom-left: fiber
lengths of and input powers to the two types of spans are exactly the
same. Bottom-right: all fiber spans are identical in length and input
signal power as well as the ordering of fibers (SMF followed by DCF).

[0027]FIG. 19 illustrates a system with OPC in the middle and having the
"one-for-many" scaled translational symmetry between RDF+SMF and SMF+RDF
spans;

[0028] FIG. 20 illustrates a typical eye diagram of optical signals
received at the end of the system with the "one-for-many" scaled
translational symmetry;

[0029]FIG. 21 illustrates a comparative system with no "one-for-many"
scaled translational symmetry but with OPC in the middle;

[0031] FIG. 23 illustrates typical received optical eye diagrams of the
two comparative systems. Top: the system with OPC in the middle; Bottom:
the system without OPC;

[0032]FIG. 24 illustrates a transmission line consisting of +NZDSFs,
-NZDSFs, and DCFs compensating the dispersion slope;

[0033]FIG. 25 illustrates received eye diagrams of the 2nd DEMUX channel.
Top row: transmission results of the setup in FIG. 24. Top-left: fiber
nonlinearity is OFF, the signal is only impaired by amplifier noise.
Top-right: fiber nonlinearity is ON, no extra penalty is visible. Bottom
row: degraded transmission results when all -NZDSFs are replaced by
+NZDSFs. Bottom-left: with OPC. Bottom-right: without OPC, of the 3rd
MUX/DEMUX channel;

[0034] FIG. 26 illustrates scalability and cascadability of the
nonlinearity-suppressed NZDSF transmission line in FIG. 24. Top: the
number of circulations on each side of OPC is doubled to ten times and
the signal power is increased by 3 dB. Bottom: two identical transmission
lines as in FIG. 24 are in cascade all-optically and the signal power is
increased by 3 dB. The eye diagrams are still of the 2nd DEMUX channel;

[0035]FIG. 27 illustrates a transmission line consisting of ten fiber
spans on each side of OPC, each span has 50 km DSF and a
slope-compensating DCF;

[0036]FIG. 28 illustrates received eye diagrams of the 2nd DEMUX channel.
Top row: transmission results of the setup in FIG. 27. Top-left: fiber
nonlinearity is OFF, the signal is only impaired by amplifier noise.
Top-right: fiber nonlinearity is ON. Bottom row: transmission results
when the setup in FIG. 27 is modified by setting D=0 ps/nm/km for the
DCFs while keeping the dispersion slope. Bottom-left: with OPC in the
middle of the link. Bottom-right: when OPC is removed;

[0037]FIG. 29 illustrates the signal power and dispersion maps for a
cascade of two fiber spans in scaled translational symmetry with scaling
ratio R=1. Top: the variation of signal power along the propagation
distance. Bottom: the dispersion map, namely, the variation of
accumulated dispersion along the propagation distance;

[0038] FIG. 30 illustrates the signal power and dispersion maps for a
cascade of two fiber spans in scaled translational symmetry with lumped
dispersion compensators. Top: the variation of signal power along the
propagation distance. Bottom: the dispersion map, namely, the variation
of accumulated dispersion along the propagation distance;

[0039]FIG. 31 illustrates a transmission line consists of 6 pairs of
fiber spans, with the first span in each pair having 50 km SMF followed
by 50 km RDF then 16 dB EDFA gain, and the second span having 40 km RDF
followed by 40 km SMF then 20 dB EDFA gain;

[0040] FIG. 32 illustrates a transmission line consists of 6 pairs of
fiber spans, with the first span in each pair having 50 km SMF followed
by 50 km RDF then 16 dB EDFA gain, and the second span having 40 km SMF
followed by 40 km RDF then 20 dB EDFA gain;

[0041] FIG. 33 illustrates the transmission results with δD=0 and
amplifier noise turned off to signify the nonlinear effects. Top:
received optical eye diagram of the scaled translationally symmetric
setup in FIG. 31. Bottom: received optical eye diagram of the setup in
FIG. 32 without scaled translational symmetry;

[0050] FIG. 42 illustrates optical eye diagrams at the end of
transmissions. Top: of a conventional design without translational
symmetry. Middle: of a system with scaled translational symmetry. Bottom:
of a system with scaled translational symmetry and mid-span SPM;

[0052] FIG. 44 illustrates a typical eye diagram of optical signals
received at the end of the optimized system with "one-for-many" scaled
translational symmetry and mid-span SPM;

[0053] FIG. 45 illustrates two portions of dispersion-compensating fiber
packaged into a compact module or cabled into a transmission line, where
the first portion may have an intentionally increased loss coefficient to
form a scaled translational symmetry with a transmission fiber, while the
second portion could have the lowest possible loss coefficient and does
not need to satisfy any scaling rule;

[0054]FIG. 46 illustrates the power map and M-type dispersion map over
the transmission distance of two traditional fiber spans;

[0055] FIG. 47 illustrates the power map and N-type dispersion map over
the transmission distance of two matched fiber spans for a scaled
translational symmetry;

[0057] FIG. 49 illustrates a test transmission system consisting of 6
recirculating loops with an M-type dispersion map on the left side, an
optical phase conjugator in the middle, then on the right side another 6
loops identical to the ones on the left. Each recirculating loop consists
of two identical spans of 100 km SMF followed by a CDCM_M, as shown on
the top of FIG. 48;

[0058]FIG. 50 illustrates a test transmission system consisting of 6
recirculating loops with an N-type dispersion map on the left side, an
optical phase conjugator in the middle, then on the right side another 6
loops identical to the ones on the left. Each recirculating loop has 100
km SMF followed by a CDCM_N, then 100 km SMF followed by a 20 dB EDFA, as
shown in the middle of FIG. 48;

[0059] FIG. 51 illustrates a test transmission system consisting of 6
recirculating loops with an optimized N-type dispersion map on the left
side, an optical phase conjugator in the middle, then on the right side
another 6 loops identical to the ones on the left. Each recirculating
loop consists of 100 km SMF followed by an ODCM, then 100 km SMF followed
by a 20 dB EDFA, as shown at the bottom of FIG. 48; and

[0060] FIG. 52 illustrates optical eye diagrams at the end of
transmissions. Top: of a conventional line with the M-type dispersion
map. Middle: of a line with conventional DCMs in the N-type dispersion
map. Bottom: of a system using optimized DCMs and scaled translational
symmetry.

DETAILED DESCRIPTION

[0061]FIG. 1 illustrates an optical communications system 102 according
to a preferred embodiment, comprising nodal elements 104, 106, 108, and
110 coupled by optical communication links 112, 114, and 116 as shown. As
used herein, optical communications link or fiber optic link refers to a
collection of optical elements including at least one optical fiber
transporting an optical signal between an optical source and an optical
receiver, while nodal element refers to an item comprising such optical
source or receiver. One example of a nodal element is a
telecommunications network node (e.g., as would be contained in a central
office) that multiplexes/demultiplexes optical signals, converts the
signals between electrical and optical form, and provides or processes
the underlying data. Another example of a nodal element is a regenerative
repeater. Typically, nodal elements are involved in electrical-to-optical
and optical-to-electrical conversion, and therefore represent expensive
hardware investments. Thus, generally speaking, it is desirable to
provide optical communications links that allow for greater distances and
higher data capacities between adjacent nodal elements. It is to be
appreciated, however, that one or more of the nodal elements 104, 106,
108, and 110 may be all-optical in nature (e.g., all-optical computing
networks, all-optical signal processors/conditioners, etc.) without
departing from the scope of the embodiments.

[0062] Optical communications link 114 comprises a first fiber span 118
and a second fiber span 120. According to an embodiment, the first and
second fiber spans 114 and 118 satisfy at least one of the scaled
translational symmetry conditions that are described further herein such
that, for an optical signal propagating from the nodal element 106 to the
nodal element 108, the second fiber span 118 at least partially
compensates for at least one nonlinearity introduced along the first
fiber span 114.

[0063] FIG. 2 illustrates the fiber span 118 according to an embodiment,
comprising a fiber line 202 and a fiber line 204, the fiber line 202
having two fiber segments 206 and 208, the fiber line 204 having a single
fiber segment 210. As used herein, fiber span refers to any contiguous
portion of an optical communications link that includes at least one
optical fiber, and may include amplifiers or other optical processing
elements positioned therealong. As used herein, fiber line refers to a
contiguous section of optical fiber positioned along a fiber span. A
fiber line may itself comprise multiple fiber segments with different
properties that are spliced together or otherwise connected to each
other.

[0064] As illustrated in FIG. 2, fiber span 118 further comprises
amplifiers 212, 214, and 216 connected in a manner that provides the
required amplifications for accommodating one or more of the embodiments
herein. In subsequent drawings and descriptions herein, it may be
presumed that amplifiers are provided at various points along a fiber
span providing the required amplifications, even where amplifiers are not
explicitly drawn. One skilled in the art would readily be able to derive
appropriate placements and parameters for such amplifiers, such as
erbium-doped fiber amplifiers (EDFAs), Raman-pumped amplifiers, etc., in
view of the present teachings, and therefore in many drawings and
descriptions herein such amplifiers might be omitted for clarity in view
of the particular context. It is to be appreciated that the configuration
of FIG. 2 represents but one example of many different possibilities for
the selection and sequencing of the various fiber lines, fiber segments,
amplifiers, and other components according to the present teachings.

[0065] By way of example, as the above terms are used herein, a typical
wide-area optical communications network having multiple nodal elements
may comprises one or more optical communications links between any two
adjacent nodal elements. A typical optical communications link may
comprise one or more fiber spans. A typical fiber span may comprise one
or more fiber lines, along with zero or more amplifiers or other optical
processing elements. A typical fiber line may consist of a single,
uniform fiber segment, or may comprise two or more fiber segments having
different properties. For example, a fiber line may comprise a "standard"
single-mode transmission fiber segment for propagating an optical signal
across a large distance connected to a dispersion-compensating fiber
segment.

[0066] Fiber lines and fiber segments may be characterized by a plurality
of propagation characteristic profiles, each propagation characteristic
profile describing the dependence of a propagation characteristic on a
distance along the direction of propagation (termed the z-direction
herein) from a reference point. One such propagation characteristic
profile is a loss/gain coefficient profile α(z) describing the
dependence of the loss-gain coefficient α on the distance along the
direction of propagation from a reference point. Another such propagation
characteristic profile is a second-order dispersion coefficient profile
β2(z) describing the dependence of the second-order dispersion
coefficient β2 on the distance along the direction of
propagation from a reference point. Another such propagation
characteristic profile is a third-order dispersion coefficient profile
β3(z) describing the dependence of the third-order dispersion
coefficient β3 on the distance along the direction of
propagation from a reference point. For many optical fibers in use today,
the α(z) and β2(z) profiles of the fiber segments are
constant along the entire length of a fiber segment. If a fiber line
contains two such adjacent fiber segments having different properties,
the α(z) and β2(z) profiles of the fiber line can be
represented by straight, horizontal plots with step-function variations
at the z-location of the intersection of the two fiber segments.
Advantageously, the embodiments herein are applicable for many different
optical fiber types having constant or spatially varying propagation
characteristics.

[0067] Other such propagation characteristic profiles include a Kerr
coefficient profile γ(z) and a Raman coefficient profile g(z). Each
of the Kerr coefficient profile γ(z) and Raman coefficient profile
g(z) represents one type of a nonlinear coefficient profile that is
referenced herein as γg(z). For clarity, it is to be
appreciated that γg(z) represents a more generalized nonlinear
coefficient profile and is not necessarily limited to the Kerr
coefficient profile γ(z). For example, in one or more of the
embodiments, γg(z) can correspond to the Raman coefficient
profile g(z) or to a different nonlinear coefficient that is important in
any particular physical context. Another propagation characteristic
profile, termed herein a nonlinear coefficient-power product profile
having a symbol (γgP)(z), comprises the product
γg(z)P(z), where P(z) represents a power profile for the
optical signal propagating down the fiber line/segment. If a particular
fiber line/segment is not yet installed in an operational optical
communications link, (γgP)(z) can be defined by using a
nominal, targeted, or otherwise computable power P(z), such as can be
yielded, for example, by assuming a value of P0 at a reference
location and computing P(z) using the known loss/gain coefficient profile
α(z).

[0068]FIG. 3 illustrates an optical communications link 302 according to
an embodiment that can be used, for example, in coupling between two
adjacent ones of the nodal elements of FIG. 1, supra. Optical
communications link 302 comprises a first fiber span 304 including a
first fiber line 310. For the embodiment of FIG. 3, the first fiber line
310 simply comprises a single first fiber segment 312 although, notably,
additional fiber segments can be included in the first fiber line 310 in
other embodiments. The first fiber line 310 comprises a first loss/gain
coefficient profile α(z), a first second-order dispersion
coefficient profile β2(z), and a first nonlinear
coefficient-power product profile (γgP)(z), where z is
measured from a reference point along the first fiber line 310 that may
be located, but is not required to be located, at an input to the first
fiber line 310. Optical communications link 302 further comprises a
second fiber span 306 including a second fiber line 314. For the
embodiment of FIG. 3, the second fiber line 314 simply comprises a single
second fiber segment 316 although, notably, additional fiber segments can
be included in the second fiber line 314 in other embodiments. The second
fiber line 314 comprises a second loss/gain coefficient profile
α'(z), a second second-order dispersion coefficient profile
β'2(z), and a second nonlinear coefficient-power product
profile (γgP)'(z), where z is measured from a reference point
along the second fiber line 314 that may be located, but is not required
to be located, at an input to the second fiber line 314.

[0069] Optical communications link 302 further comprises an optical phase
conjugator 308 optically coupled between the first fiber span 304 and the
second fiber span 306, an optical signal received by the first fiber span
304 being propagated respectively through the first fiber span 304, the
optical phase conjugator 308, and the second fiber span 306. Preferably,
the first fiber span, the second fiber span, and the optical phase
conjugator are configured such that, for locations lying along the first
fiber segment 312 and the second fiber segment 316, α'(z) is in
substantial scaled translational symmetry with α'(z) by a first
constant R, β'2(z) is in substantial scaled translational
symmetry with β2(z) oppositely signed (that is, with
-β2(z)) by the first constant R, and (γgP)'(z) is in
substantial scaled translational symmetry with (γgP)(z) by the
first constant R, whereby progressive compensation along the second fiber
segment 316 of at least one nonlinearity introduced along the first fiber
segment 312 is facilitated.

[0070] As indicated by the plots in FIG. 3, substantial scaled
translational symmetry characterizes a relationship between a first
profile and a second profile if the first profile, when expanded along a
first axis by a scaling factor R and contracted along a second axis by
the scaling factor R, is in substantial correspondence with said second
profile. In the embodiment of FIG. 3, the first axis is the ordinate and
the second axis is the abscissa of the profiles shown, although the scope
of the present teachings is not necessarily so limited. Stated in
algebraic vector form, the conditions illustrated in the plots of FIG. 3
can be expressed as R[α(Rz), β2(Rz),
(γgP)(Rz)]≈[α'(z), β'2(z),
(γgP)'(z)].

[0071] The substantial correspondence between the scaled (and,
effectively, translated) plots is generally more important near the
inputs where (γgP)(z) and (γgP)'(z) are relatively
high, and generally less important later on where (γgP)(z) and
(γgP)'(z) are relatively low. The particular degrees of
correspondence between the scaled and translated plots required for
sufficient facilitation of nonlinear effect compensation would be readily
determinable, whether empirically, by simulation, or by derivation, by a
person skilled in the art without undue experimentation in view of the
present disclosure.

[0072] In one embodiment the scaling factor R lies between about 0.05 and
20. In another embodiment, the scaling factor R lies between about 0.1
and 0.2 or between about 5 and 10. As indicated by the above-referenced
value ranges for R, which are presented only by way of example and not by
way of limitation, the length ratios between the first fiber segment 312
and the second fiber segment 316 can vary widely without departing from
the scope of the embodiments. For R>1, the first fiber segment 312 is
longer than the second fiber segment 316, while for R<1, the first
fiber segment 312 is shorter than the second fiber segment 316. In other
embodiments, the optical phase conjugator 308 can be omitted in
conjunction with providing for certain optical signal phase relationships
as described further hereinbelow. In one embodiment, the optical
communications link 302 is configured such that a complex amplitude of
the optical signal at an input to the second fiber line 314 is
proportional to a complex conjugate of the complex amplitude of the
optical signal at an input to the first fiber line 310.

[0073] In one embodiment, the first and second fiber lines 310 and 314
further comprise third-order dispersion coefficient profiles
β3(z) and β'3(z) along that are in substantial scaled
translational symmetry along the first and second segments 312 and 316.
For this embodiment, each of the first and second fiber segments 312 and
316 may comprise a non-zero dispersion-shifted fiber (NZDSF) or,
alternatively, may comprise a dispersion-shifted fiber (DSF).

[0074] In another embodiment, the constraint in which all three of
α(z), β(z) (oppositely signed), and (γP)(z) are in
substantial scaled symmetry is at least partially relaxed. For this
embodiment, the first and second fiber lines 310 and 314 are in
substantial scaled translational symmetry by a common scaling factor R
with respect to β(z) (oppositely signed) and with respect to one or
both of α(z) and (γP)(z) along the first and second fiber
segments 312 and 316 for facilitating progressive compensation along the
second fiber segment 316 of at least one nonlinearity introduced along
the first fiber segment 312.

[0075] FIG. 4 illustrates the optical communications link 302 of FIG. 3 as
redrawn with simplified notation for clarity in describing further
embodiments herein. In addition to removing amplifiers for clarity, the
fiber span/line/segment notations have been replaced by a simple
lettering scheme in which a contiguous length of optical fiber is denoted
by a loop and identified by a letter. Moreover, the presence of
substantial scaled translational symmetry between two such contiguous
lengths of optical fiber by a common scaling factor, whether it be with
respect to all three of α(z), β(z) (oppositely signed), and
(γgP)(z), or whether it be with respect to β(z)
(oppositely signed) and one or both of α(z) and
(γgP)(z), is denoted by prime symbol relationships (e.g., A
and A'), and, without loss of generality, the acronymed phrase "ST
symmetry" is used to identify such condition between A and A'. Finally,
without loss of generality, the contiguous lengths of optical fibers A
and A' are each referred to as fiber spans, it being understood that they
are particular cases from the broader definition of fiber span supra
applicable when a fiber span consists of a single fiber line that
consists of a single fiber segment.

[0076]FIG. 5 illustrates an optical communications link 502 according to
an alternative embodiment to that of FIGS. 3 and 4 having the optical
phase conjugator omitted. FIG. 6 illustrates an optical communications
link 602 with enhanced notations in which a larger double-looped symbol
represents a transmission fiber (TF) span (e.g., that would extend over
tens or hundreds of kilometers) and in which a smaller single-looped
symbol represents a dispersion-compensating fiber (DCF). Optical
communications link 602 comprises a TF span A in ST symmetry a TF span
A', and a DCF span B in ST symmetry with a DCF span B'. The DCF span B is
designed for compensating the dispersion accumulated in TF span A, while
the DCF span B' is designed for compensating the dispersion accumulated
in TF span A'.

[0077] Thus, advantageously, the fiber spans A, B, A', and B' are in a
beneficial cross-paired relationship that simultaneously and efficiently
achieves multiple goals. In particular, the TF span A is paired with DCF
span B for dispersion compensation while at the same time being paired
with TF span A' for nonlinearity compensation. Likewise, the TF span A'
is paired with DCF span B' for dispersion compensation while at the same
time being paired with TF span A for nonlinearity compensation.

[0078] For clarity of presentation, the notations of FIGS. 7A-7O
hereinbelow follow the notational schemes of FIG. 6. FIGS. 7A-7O
illustrate, by way of example and not by way of limitation, some of the
many advantageous ways that fiber spans in STS symmetry may be arranged
in accordance with the present teachings, where boxes labeled OPC are
optical phase conjugators and dotted-line boxes represent common
dispersion-compensating module packages (or such packages co-located with
an OPC), and where optical signals propagate from the left side to the
right side of the page.

[0079] Referring to FIGS. 7A-7O, in one embodiment, each of the TF spans
A, A'. C. and C' has a constant α(z) value that is less than 0.5,
and each of the DCF spans B, B', D, and D' has a constant α(z)
value that is greater than 1.0. The scaling factors associated with the
different STS symmetry pairs can be different or they can be the same.

[0080] FIG. 8 illustrates, with reference back to the nomenclature and
notational schemes of FIGS. 3-5, supra, an optical communications link
802 according to an embodiment comprising a fiber span 804 having a fiber
line 806, the fiber line 806 having fiber segments 808 and 810. The
optical communications link 802 further comprises an optical phase
conjugator (that is omitted in other embodiments). The optical
communications link 802 further comprises a fiber span 812 having a fiber
line 814, the fiber line 814 having fiber segments 816 and 818 positioned
as shown. For this embodiment, the fiber segments 808 and 816 are in
substantial scaled translational symmetry, whether it be with respect to
all three of α(z), β(z) (oppositely signed), and
(γgP)(z), or whether it be with respect to β(z)
(oppositely signed) and one or both of α(z) and
(γgP)(z), but the fiber segments 810 and 818 do not satisfy
such condition. However, the fiber segments 810 and 818 are connected to
fiber segments 808 and 816 at "z" locations such that (γgP)(z)
is generally small compared to a maximum value thereof. This
non-satisfaction does not substantially confound the progressive
compensation in fiber line 814 of nonlinearities introduced in fiber line
806 because the power levels in the fiber segments 810 and 818 are
relatively low and nonlinearities are therefore relatively small.
However, this non-satisfaction is a constraint relaxation that
accommodates length variations in the fiber lines 810 and 818 such that
other system design goals including dispersion compensation can be
properly addressed. In one embodiment, the fiber segments 810 and 818 are
connected to fiber segments 808 and 816 at "z" locations where
(γgP)(z) is less than 10% of its maximum value along fiber
segments 808 and 816.

[0081] A set of further embodiments is now described with respect to FIGS.
7A-7O and FIG. 8. For these embodiments, any combination of the ST
symmetry pairs A-A', B-B', C-C', and D-D' in FIGS. 7A-7O can be replaced
by a pair of spans similar to the spans 804 and 806 of FIG. 8 in which
only the first fiber segments of the spans (e.g., 808 and 816) meet the
ST symmetry condition while the second fiber segments of the spans (e.g.,
810 and 818 do not meet an ST symmetry condition, provided that the
second fiber segments connect to the first fiber segments where the value
of (γgP)(z) is generally small compared to a maximum value
thereof. Thus, for example, letting X represent the fiber segment 810 and
Y' represent the fiber segment 818, the ST symmetry pair C-C' in FIG. 7N
can become the pair (C & X)-(C' & Y') according to these embodiments (and
any combinations of C-C' and the other ST symmetry pairs) without
confounding the nonlinearity compensation thereof. Benefits similar to
those described with respect to FIG. 8 supra are advantageously achieved.

[0082]FIG. 9 illustrates an optical communications link 902 for guiding a
wavelength-division multiplexed (WDM) optical signal between a first
nodal element, or node, 904 and a second node 906, the optical signal
having a plurality of channels. The optical communications link 902
comprises a first fiber span 908 receiving the optical signal from the
first node 904 and having Kerr nonlinear effects introducing ghost-pulse
artifacts into at least one of the channels. The first fiber span 908 is
dispersion-compensated. The optical communications link 902 further
comprises a second fiber span 910 transferring the optical signal to the
second node 906 and having Kerr nonlinear effects similar to those of the
first fiber span introducing similar ghost-pulse artifacts into the at
least one channel. The optical communications link 902 further comprises
a self-phase modulating device 912 positioned between the first and
second fiber spans 908 and 910, comprising an array 914 of self-phase
modulators corresponding respectively to each of the plurality of
channels configured such that the ghost-pulse artifacts introduced by the
first fiber span 908 are substantially reduced upon arrival of the
optical signal at the second node 906. In one embodiment, each of the
self-phase modulators introduces an approximately 180-degree phase shift
at a nominal power level of pulse peaks of the associated channel. The
self-phase modulating device further comprises a WDM demultiplexer 916
coupling the first fiber span 908 to the self-phase modulators and a WDM
multiplexer 918 coupling the self-phase modulators to the second fiber
span. The self-phase modulators can comprise Kerr optical fibers,
nonlinear lithium niobate waveguides, and/or semiconductor optical
amplifiers. Preferably, the self-phase modulators each have a dispersion
characteristic designed to limit spectral broadening of the
phase-modulated pulses in the associated channel.

[0083] Whereas many alterations and modifications of the present invention
will no doubt become apparent to a person of ordinary skill in the art
after having read the descriptions herein, it is to be understood that
the particular embodiments shown and described by way of illustration are
in no way intended to be considered limiting. Therefore, reference to the
details of the embodiments are not intended to limit their scope, which
is limited only by the scope of the claims set forth below.

[0084] Group-velocity dispersion and optical nonlinearity are the major
limiting factors in high-speed long-distance fiber-optic transmissions
[1, 2]. Dispersion-compensating fibers (DCFs) have been developed to
offset the dispersion effects of transmission fibers over a wide
frequency band. The most advanced DCFs are even capable of slope-matching
compensation, namely, compensating the dispersion and the dispersion
slope of the transmission fiber simultaneously [3, 4]. Nevertheless, DCFs
could hardly be designed and fabricated to match exactly the dispersion
and the slope of transmission fibers simultaneously. In general, it is
difficult to perfectly compensate the fiber dispersion across a wide
frequency band. There are always residual dispersion and higher order
derivatives, even using the best slope-matching DCFs [5, 6, 7]. The
significance of the residual dispersions increases as the total signal
bandwidth becomes wider [8]. It has been proposed for some time that
optical phase conjugation (OPC) may be employed in the middle of a
transmission line to equalize the dispersion effect of the transmission
fibers [9]. Furthermore, theoretical and experimental studies have proved
the feasibility of using OPC to compensate the fiber nonlinearities, at
least partially [10, 11, 12]. In the past, the application of OPC has
been limited by the lack of performing conjugators that require low pump
powers, operate over wide bandwidths, and suffer low penalties. Such
technical difficulties and the inability of compensating the dispersion
slope have been to OPC's disadvantage in competing with DCFs as
dispersion compensators. However, it is noted that the performance of
optical phase conjugators has recently been and will continue to be
improved significantly [13, 14]. Moreover, we argue that OPC and modern
DCFs may work together nicely to complement each other's functionalities.
On one hand, transmission fibers and DCFs may be combined into fiber
spans with zero dispersion slope, then OPC is able to equalize the
residual dispersion and the slope of dispersion slope among such spans.
On the other hand, the flexible designs and various choices in the
dispersion parameters of specialty fibers, in particular DCFs, make it
possible to construct fiber trans-mission lines that manifest "scaled
symmetries" about the OPC, which are desired properties to effectively
suppress fiber nonlinearities [15, 16, 17].

[0085] Based on the nonlinear Schrodinger equation (NLSE), it has been
shown that OPC enables one fiber transmission line to propagate inversely
(thus to restore) an optical signal that is nonlinearly distorted by the
other, when the two fiber lines are mirror-symmetric about the OPC in the
scaled sense [11, 15, 17]. Preliminary experiments have confirmed such
effect of nonlinear compensation [11, 12]. Unfortunately, the mirror
symmetry requires that the conjugating fiber segments have opposite
loss/gain coefficients, the same sign for the second-order dispersions,
and opposite third-order dispersions. These conditions are not
conveniently fulfilled in many practical fiber transmission systems. In
particular, a mirror-symmetric signal power profile is possible only when
some transmission fibers are made distributively amplifying by means of
distributed Raman pumping [18] or using distributed Er-doped fiber
amplifiers (EDFAs) [19], so to obtain a constant net gain in
correspondence to the loss coefficient of other fibers, or all fibers are
rendered lossless. Recent experiments [20, 21, 22] have indeed
demonstrated near constant-power or low power-excursion optical
transmissions. However, there are still concerns of cost, reliability,
and double-Rayleigh-scattering noise with distributive Raman
amplification [18]. For any distributive amplifier, the loss of pump
power makes it difficult to maintain a constant gain in a long
transmission fiber. Consequently, the mismatch in signal power profiles
degrades the result of nonlinear compensation. Yet another shortcoming of
the previous schemes [11, 12] is that they do not compensate higher-order
dispersions, which could turn into a significant limitation in wide-band
transmission systems. By contrast, a recently proposed method of
nonlinearity compensation using scaled translational symmetry requires
that the conjugating fiber segments have the same sign for the loss/gain
coefficients, opposite second-order dispersions, and the same sign for
the third-order dispersions [16, 17]. Such conditions are naturally
satisfied, at least approximately, in conventional fiber transmission
systems, where, for example, a standard single-mode fiber (SMF) may be
paired with a DCF as conjugating counterparts. In Refs. [16, 17], we have
briefly touched upon the basic idea and feasibility of nonlinearity
compensation using scaled translational symmetry. In this paper, we shall
present an extensive and systematic study of the theory and practical
applications of scaled translational symmetry in fiber transmission
systems for nonlinearity compensation. Most importantly, we demonstrate
that the combination of scaled nonlinearity, translational symmetry, OPC,
and slope-matching dispersion compensation makes our proposals of
nonlinearity compensation rather practical and highly performing. The
notion of scaling fiber nonlinearity is not entirely new. The concept was
proposed and utilized by Watanabe et al. in their 1996 paper [11], which
however was limited to the mirror-symmetric configuration, and presented
embodiments using segmented fibers which might not be convenient to
implement in practice. Even though we may be the first to emphasize the
concept and importance of scaled translational symmetry to nonlinearity
compensation in fiber transmission lines [16, 17], it was noted
previously by Marhic et al. [23] that two fibers having opposite
dispersions and with OPC in the middle may compensate each other's Kerr
nonlinear effects. However, Ref. [23] did not discuss any practical
embodiment, nor did it mention the scaling of nonlinearity which is
indispensable for practically implementing translationally symmetric
transmission lines. Both Refs. [11, 23] had the effect of
dispersion-slope neglected, and did not worry about the Raman effect
among wavelength-division multiplexed (WDM) channels. By contrast, this
present paper strives for the most generality, and it might be one of the
early proposals for optimizing fiber transmission systems by combining
the necessary and available four elements, namely, scaled nonlinearity,
translational symmetry, OPC, and slope-matching dispersion compensation.
It is this combination that signifies the present work and makes our
proposals of nonlinearity compensation rather practical and highly
performing. Furthermore, it is found that even without OPC, the
combination of the remaining three elements could still significantly
improve the performance of fiber transmission lines. Two fiber spans in a
scaled translational symmetry may cancel out their intra-channel
nonlinear effects to a large extent, and a significant reduction of
intra-channel nonlinear effects may be achieved in a long-distance
transmission line consisting of multiple pairs of scaled translationally
symmetric spans.

Basics of Dispersive and Nonlinear Wave Propagation in Fibers

[0086] The eigenvalue solution of Maxwell's equations in a single-mode
fiber determines its trans-verse model function and propagation constant
β(ω) as a function of the optical frequency ω[24, 25].
When a fiber transmission line is heterogeneous along its length, the
propagation constant could also depend on the longitudinal position z in
the line, and may be denoted as β(z, ω). The slow-varying
envelope form,

E(z,t)=A(z,t)exp[i∫zβ0(ζ)dζ-iω0-
t], (1)

with β0(z)defβ(ω0, z), is often employed to
represent an optical signal, which may be of a single time-division
multiplexed channel or a superposition of multiple WDM channels. The
evolution of the envelope A(z, t) in an optical fiber of length L is
governed by the nonlinear Schrodinger equation (NLSE) [17, 25],

are the z-dependent dispersion coefficients of various orders [26],
γ(z) is the Kerr nonlinear coefficient of the fiber, g(z, t) is the
impulse response of the Raman gain spectrum, and denotes the convolution
operation [17]. Note that all fiber parameters are allowed to be
z-dependent, that is, they may vary along the length of the fiber.
Because of the definition in terms of derivatives, β2 may be
called the second-order dispersion (often simply dispersion in short),
while β3 may be called the third-order dispersion, so on and so
forth. The engineering community has used the term dispersion for the
parameter D=dvg-1/dλ, namely, the derivative of the
inverse of group-velocity with respect to the optical wavelength, and
dispersion slope for S=dD/dλ[1]. Although β2 and D are
directly proportional to each other, the relationship between
β3 and S is more complicated. To avoid confusion, this paper
adopts the convention that dispersion and second-order dispersion are
synonyms for the β2 parameter, while dispersion slope and
third-order dispersion refer to the same β3 parameter, and
similarly the slope of dispersion slope is the same thing as the
fourth-order dispersion β4.

[0087] Had there been no nonlinearity, namely γ(z)=g(z, t)≡0,
equation (2) would reduce to,

which could be solved analytically using, for example, the method of
Fourier transform. Let F denote the linear operator of Fourier transform,
a signal A(z, t) in the time domain can be represented equivalently in
the frequency domain by,

(z,ω)F
A(z,t)=∫A(z,t)exp(iωt)dt=∫E(z,t)exp[i(ω0+.omeg-
a.)t]dt. (5)

Through a linear fiber, a signal (z1, ω) at z=z1 would
be transformed into (z2, ω)=H(z1, z2, ω)
(z1, ω) at z2≧z1, where the transfer function
H(z1, z2, ω) is defined as,

In the time domain, the signals are related linearly as A(z2,
t)=P(z1, z2)A(z1, t), with the linear operator P(z1,
z2) given by,

P(z1,z2)F-1H(z1,z2,ω)F. (7)

Namely, P(z1, z2) is the concatenation of three linear
operations: firstly Fourier transform is applied to convert a temporal
signal into a frequency signal, which is then multiplied by the transfer
function H(z1, z2, ω), finally the resulted signal is
inverse Fourier transformed back into the time domain. In terms of the
impulse response,

h(z1,z2,t)F-1[H(z1,z2,ω)], (8)

P(z1, z2) may also be represented as,

P(z1,z2)=h(z1,z2,t), (9)

where denotes functional convolution. That is, the action of P(z1,
z2) on a time-dependent function is to convolve the function with
the impulse response. All linear operators P(z1, z2) with
z1≦z2, also known as propagators, form a semigroup [27]
for the linear evolution governed by equation (4).

[0088] However, the existence of nonlinear terms in equation (2) makes the
equation much more difficult to solve. Fortunately, when the signal power
is not very high so that the nonlinearity is weak and may be treated as
perturbation, the output from a nonlinear fiber line may be represented
by a linearly dispersed version of the input, plus nonlinear distortions
expanded in power series of the nonlinear coefficients [28]. In practical
transmission lines, although the end-to-end response of a long link may
be highly nonlinear due to the accumulation of nonlinearity through many
fiber spans, the nonlinear perturbation terms of higher orders than the
first are usually negligibly small within each fiber span. Up to the
first-order perturbation, the signal A(z2, t) as a result of
nonlinear propagation of a signal A(z1, t) from z1 to
z2≧z1, may be approximated using,

where A(z2, t)≈A0(z2, t) amounts to the
zeroth-order approximation which neglects the fiber nonlinearity
completely, whereas the result of first-order approximation A(z2,
t)≈A0(z2, t)+A1(z2, t) accounts in addition
for the lowest-order nonlinear products integrated over the fiber length.
The term A1(•, t) is called the first-order perturbation
because it is linearly proportional to the nonlinear coefficients
γ(•) and g(•, t).

Principles of Dispersion and Nonlinearity Compensation Using OPC

[0089] Dispersion equalization by OPC may be explained nicely using
transfer functions in the frequency domain [29]. Optical signals at a
fixed position in a fiber, possibly of many channels wavelength-division
multiplexed together, may be described by a total electrical field
E(t)=A(t) exp(-iω0t), with the position parameter omitted. The
signals are fully represented by the slow-varying envelope A(t), or
equivalently, by the Fourier transform of the envelope (ω)=FA(t).
Leaving aside the loss/gain and neglecting the nonlinearities, the linear
dispersive effect of a fiber transmission line is described by a
multiplicative transfer function,

being the dispersions accumulated along the fiber length, and the
dispersion parameters {βk}k≧2 being defined as in
equation (3). A fiber line with such dispersion parameters transforms a
signal (ω) into H(ω) (ω), while OPC acts as a linear
operator that changes the same signal into OPC[ (ω)]= *(-ω).
Consider two fiber transmission lines that are not necessarily identical,
but nevertheless have accumulated dispersions satisfying the conditions,

bkR=(-1)kbkL,.A-inverted.k≧2, (14)

so that HR(ω)=HL(-ω), where the super- and
sub-scripts L, R are used to distinguish the two fiber lines on the left
and right respectively. When OPC is performed in the middle of the two
fiber lines, the entire setup transforms an input signal (ω) into,

HR(ω)OPC[HL(ω)
(ω)]=HR(ω)HL*(-ω) *(-ω)= *(-ω).
(15)

If (ω) is the Fourier transform of A(t), then the output signal
*(-ω) corresponds to A*(t) in the time domain, which is an
undistorted replica of the input signal A(t) up to complex conjugation.
This proves that the dispersion of a transmission line with OPC in the
middle may be compensated over a wide bandwidth, when the dispersion
coefficients of the odd orders on the two sides of OPC, b2k+1L
and b2k+R with k≧1, in particular the third-order
dispersions b3L and b3R, are both compensated to
zero, or they are exactly opposite to each other, while the even-order
dispersion coefficients are the same on both sides. If a link has
b3R=-b3L, or even b3R=b3L=0, then
it is compensated at least up to and including the fourth-order
dispersion b4. It is worth pointing out that the center frequency of
the signal band may be shifted by the OPC from ω0L on the
left side to ω0R on the right side,
ω0L≠ω0R, and the dispersion
parameters on the two sides of OPC are defined with respect to the
corresponding center frequencies.

[0090] To compensate the nonlinearity of transmission fibers, our method
of using scaled translational symmetry [16, 17] requires that the
conjugating fiber segments have the same sign for the loss/gain
coefficients, opposite second-order dispersions, and the same sign for
the third-order dispersions. Such conditions are naturally satisfied, at
least approximately, in conventional fiber transmission systems, where,
for example, an SMF may be paired with a DCF as conjugating counterparts.
The symmetry is in the scaled sense, because the lengths of the fibers
and the corresponding fiber parameters, including the fiber loss
coefficients and dispersions, as well as the Kerr and Raman nonlinear
coefficients, are all in proportion, and the proportional ratio may not
be 1. The symmetry is translational, because the curves of signal power
variation along the fiber keep the similar shape, albeit scaled, when
translated from the left to the right side of OPC, as depicted in FIG.
10, so do the curves of any above-mentioned fiber parameter if plotted
against the fiber length. The fundamental discovery is that two fiber
lines translationally symmetric about the OPC are able to cancel each
other's nonlinearities up to the first-order perturbation. To understand
the principle, imagine two fiber lines with opposite nonlinear
coefficients but identical linear parameters of dispersion and loss/gain.
It turns out that the nonlinear effects of the two are compensated up to
the first-order perturbation, when they are used in cascade as shown in
FIG. 11. The first fiber stretching from z=-L to z=0 is a real, physical
one with parameters α(z), {βk(z)}k≧2,
γ(z), g(z, •), so that the signal propagation in which is
governed by,

-L≦z≦0. The other is a fictitious fiber stretching from z=0
to z=L, with parameters α'(z), {βk'(z)}k≧2,
γ'(z), g'(z, •) satisfying,

a'(z)=α(z-L), (17)

βk'(z)=βk(z-L),.A-inverted.k≧2, (18)

γ'(z)=-γ(z-L), (19)

g'(z,t)=-g(z-L,t),.A-inverted.tε(-∞,+∞), (20)

.A-inverted.zε[0, L]. Note that the fictitious fiber may be
unphysical because of the oppositely signed nonlinear coefficients
γ' and g' [30]. The signal propagation in this fictitious fiber
obeys the following NLSE,

0≦z≦L. FIG. 12 shows the signal power and dispersion maps
in the series of two fiber lines. It is obvious from equations (6-11) and
(17-20) that the two fiber lines would induce opposite first-order
nonlinear distortions to otherwise the same linear signal propagation
(zeroth-order approximation), because the two linear propagators
P(z1-L, z2-L) and P(z1, z2) are exactly the same, for
all z1ε[0, L] and all z2ε[z1, L], while
the Kerr nonlinear coefficients γ(z-L) and γ'(z), as well as
the Raman coefficients g(z-L, •) and g'(z, •), are exactly
opposite-valued, for all zε[0, L]. If the overall dispersion of
each fiber line is compensated to zero and the signal loss is made up by
linear optical amplifiers, then the same perturbation argument may be
applied to the two lines in cascade to show that the fiber nonlinearity
is annihilated up to the first-order perturbation. The problem is that an
optical fiber with negative nonlinear coefficients may be only
fictitious. It does not exist naturally.

[0091] For a fictitious fiber of length L and with parameters as those in
equation (21), the Kerr nonlinear coefficient γ' is
negative-valued, and the Raman gain g is reversed, or called "negative"
as well [30], in the sense that it induces optical power flow from lower
to higher frequencies, which obviously will not happen normally.
Fortunately, such fictitious fiber may be simulated by an ordinary fiber
with the help of OPC, as depicted in FIG. 13. An ordinary fiber of length
L/R may be found with parameters α'',
{βk''}k≧2, γ'', g'' satisfying the following
rules of scaling,

α''(z)=Rα'(Rz), (22)

βk''(z)=(-1)k-1Rβk'(Rz),.A-inverted.k≧2,
(23)

γ''(z)=-Qγ'(Rz), (24)

g''(z,t)=-Qg'(Rz,t),.A-inverted.tε(-∞,+∞), (25)

.A-inverted.zε[0, L/R], where R>0, Q>0 are scaling factors.
In this ordinary fiber, the NLSE of signal propagation is,

with θεR being an arbitrary phase, then a change of
variable Rz→z, and finally taking the complex conjugate of the
whole equation, equation (27) becomes mathematically identical to
equation (21). Equation (28) is actually the scaling rule for the signal
amplitudes. The physical implication is that, if a signal A'(0, t) is
injected into the fictitious fiber and the complex conjugate signal
eiθ(R/Q)1/2[A'(0, t)]* is fed to the ordinary fiber, then
the signal at any point zε[0, L/R] in the ordinary fiber is
eiθ(R/Q)1/2[A' (Rz, t)]*, which is
eiθ(R/Q)1/2 times the complex conjugate of the signal at
the scaled position Rz in the fictitious fiber. In particular, the output
signals are A'(L, t) and eiθ(R/Q)1/2[A'(L, t)]* from the
fictitious and the ordinary fibers respectively. Except for scaling the
signal power by a factor R/Q, the ordinary fiber with two phase
conjugators installed at its two ends performs exactly the same
dispersive and nonlinear signal transformation as the fictitious fiber.
Such equivalence is illustrated in FIG. 13. In practice, the phase
conjugator at the output end of the ordinary fiber may be omitted, as
most applications would not differentiate between a signal and its
complex conjugate. Replacing the fictitious fiber with negative
nonlinearities in FIG. 11 by such scaled ordinary fiber with OPC attached
at the input end, one arrives at a nonlinearity-compensating setup using
all physical components/devices: an optical phase conjugator in the
middle, an ordinary fiber on the left side stretching from z=-L to z=0
with parameters α''(z), {βk(z)}k≧2,
γ(z), g(z, •), and an ordinary fiber on the right side
stretching from z=0 to z=L/R with parameters α''(z),
{βk''(z)}k≧2, γ''(z), g''(z, •). It
follows from equations (17-20) and (22-25) that the parameters of the two
fibers are related as,

α''(z)=Rα(Rz-L), (29)

βk''(z)=(-1)k-1Rβk(Rz-L),.A-inverted.k≧2-
, (30)

γ''(z)=Qγ(Rz-L), (31)

g''(z,t)=Qg(Rz-L,t),.A-inverted.tε(-∞,+∞), (32)

.A-inverted.zε[0, L/R]. Equations (29-32) are called the scaling
rules for two fibers to form a translational symmetry in the scaled sense
about an optical phase conjugator [16, 17]. In order for two fiber lines
in scaled translational symmetry to compensate their nonlinearities up to
the first-order perturbation, it is further required that the input
signals A(-L, t) and A''(0, t) at the beginning of the two fiber lines
satisfy the following,

A''(0,t)=eiθ(R/Q)1/2[A(-L,t)]*, (33)

where θεR is an arbitrary phase. Equation (33) may be
regarded as the scaling rule for the input signals to the fibers.

[0092] The analysis has convinced us that OPC may help to compensate fiber
nonlinearities between two transmission lines that are in scaled
translational symmetry. It should be emphasized that the fiber line on
each side of OPC does not necessarily consist of only one fiber span, and
the signal intensity does not have to evolve monotonically either. The
simple setup used above should only be regarded as an example for
illustration and mathematical convenience. The proposed method of
nonlinear compensation works fine when each side of the OPC consists of
multiple fiber spans with optical amplifiers in between repeating the
signal power. In which case, each fiber on one side should be paired with
a scaled translationally symmetric counterpart on the other side, with
the parameters and input signals of the fiber pair satisfying the similar
scaling rules as in equations (29-33). Because most fibers do not start
or end at z=0 in a transmission line consisting of many spans, the
scaling rules for them would be similar to equations (29-33) but with the
position coordinates suitably adjusted. Furthermore, the scaling ratios
may vary from one pair of fibers to another. Put in words, the scaling
rules for scaled translational symmetries between pairs of fiber segments
require that each pair of fiber segments have the same sign for the
loss/gain coefficients, opposite second-order dispersions, the same sign
for the third-order dispersions, and the same positive-valued nonlinear
coefficients [30]. Moreover, a fiber may have its linear parameters
scaled by a common factor and its nonlinear coefficients scaled by
another factor, then the length of the fiber may be scaled inversely
proportional to the linear parameters, and the signal power may be
adjusted accordingly to yield the same strength of nonlinear
interactions. The conditions of "the same sign for loss coefficients and
opposite signs for the second-order dispersions" are naturally satisfied
by the transmission fibers and DCFs used in conventional transmission
systems. Another fact, simple but crucially important for practical
applications, is that nonlinear effects are significant only in portions
of fibers where the signal power is high. When scaling fiber parameters
and signal amplitudes to have two fiber spans inducing the same or
compensating nonlinear effects, it is only necessary to make sure that
the scaling rules of equations (29-32) and (33) are fulfilled in portions
of fibers experiencing high levels of signal power. Elsewhere, the
scaling rules may be loosened or neglected when the signal power is low.
Relaxing the scaling rules in portions of fibers carrying low-power
signals makes it much easier to find practical and commercially available
fibers with suitable dispersion characteristics to manage the accumulated
dispersions of individual spans.

[0093] With such scaling of nonlinearities [16, 17], both the Kerr and
Raman nonlinearities may be suppressed simultaneously if a proportional
relation is maintained between the γ and g parameters as in the
scaling rules of equations (31) and (32). When equations (31) and (32)
can not be fulfilled simultaneously, either the Kerr or the Raman
nonlinearity may be primarily targeted for compensation depending upon
the actual application. For a translational symmetry between two fibers
with opposite dispersions, the scaling rule of equation (29) requires the
same sign for the loss/gain coefficients of the two fibers, which is a
convenient condition to meet by the natural fiber losses. This is in
contrast to the mirror symmetry between two fiber segments that requires
an amplifying segment correspond to a lossy one and vice versa. Fibers
may be designed and fabricated with the requirements of scaled symmetry
taken into consideration. For a given piece of fiber, the loss
coefficient may need to be intentionally increased to meet the scaling
rule. The extra loss may be induced by, for example, doping the fiber
preform with erbium, or transition metals, or other impurities [32, 33],
macro-bending [24] the fiber or writing long-period Bragg gratings into
the fiber for scattering losses. Macro-bending may be built in a lumped
fiber module having the fiber coiled tightly with a suitable radius. Also
discrete fiber coils or Bragg gratings for light attenuation may be
implemented periodically along the length of a fiber to approximate a
continuous uniform loss coefficient. More sophisticatedly, Raman pumps
may be employed to induce gain or loss to the optical signals depending
upon the pump frequencies being higher or lower than the signal band, so
to alter the effective gain/loss coefficient of the fiber. Even though it
is rather difficult to change the dispersion of a given fiber, OPC is
capable of shifting the center frequency of the signal band, which can
fine-tune the effective dispersion at the center of the signal band, so
long as the fiber has a non-zero dispersion slope. Even though most
fibers are made of similar materials with similar nonlinear
susceptibilities, their guided-wave nonlinear coefficients measured in
W-1km-1 could be quite different due to the wide variation of
modal sizes. Unless the ratio of nonlinear coefficients matches the ratio
of dispersions, the signal powers in two conjugate fibers may have to
differ by several dB as required by the scaling rule of equation (33) for
scaled translational symmetry. Alternatively, by taking advantage of the
additivity of the first-order nonlinear perturbations, it is possible to
adjust the signal powers in different fiber spans only slightly, such
that one span of a highly-nonlinear type may compensate several fiber
spans of another type with weaker nonlinearity. This method may be called
"one-for-many" (in terms of fiber spans) nonlinearity compensation.

[0094] It should be noted that the suitability of compensating
nonlinearities among lossy fibers does not exclude the method of
translational symmetry from applying to systems with amplifying fibers
due to Raman pumping [18, 20, 21, 22] or rare-earth-element doping [19].
The scaled translationally symmetric method applies to these systems
equally well, provided that an amplifying fiber is brought into
translational symmetry with respect to another fiber with gain. In fact,
if two fibers with their intrinsic loss coefficients satisfying the
scaling rule of equation (29), then the power of the Raman pumps (forward
or backward) to them may be adjusted properly to yield effective
gain/loss coefficients satisfying the same rule of equation (29). In
particular, Raman pumped DCFs [34, 35, 36] may be conveniently tuned
translationally symmetric to a Raman pumped transmission fiber. For
systems suffering considerable nonlinear penalties originated from long
EDFAs [37], the penalties may be largely suppressed by arranging the
amplifiers into conjugate pairs with scaled translational symmetry about
the OPC. The nonlinear and gain coefficients as well as the signal
amplitudes in the amplifying fibers should obey the scaling rules. If the
dispersions of the amplifying fibers are not negligible, they should be
designed to satisfy the scaling rules as well. Finally, it is also
necessary to note the limitation of nonlinearity compensation using
scaled translational symmetry. That is, the method can only compensate
the first-order nonlinear interactions among the optical signals. The
higher-order nonlinear products are not compensated, nor is the nonlinear
mixing between transmitted signals and amplifier noise. The accumulation
of uncompensated higher-order nonlinearities and nonlinear signal-noise
mixing would eventually upper-bound the amount of signal power permitted
in the transmission fibers, so to limit the obtainable signal-to-noise
ratio, and ultimately limit the product of data capacity and transmission
distance.

Optimal Setups of Fiber-Optic Transmission Lines

[0095] Having established the basic principles of dispersion equalization
and nonlinearity compensation using OPC and scaled translational
symmetry, we shall now discuss practical designs of fiber systems for
long-distance transmissions, with realistic (commercially available) DCFs
and transmission fibers that are optimally configured according to the
basic principles of simultaneous compensation of dispersion and
nonlinearity. A long-distance trans-mission line may consist of many
fiber spans, each of which may have transmission and
dispersion-compensating fibers. Two fibers with opposite (second-order)
dispersions may be tuned translationally symmetric to each other about a
phase conjugator. For optimal non-linearity compensation, the fiber
parameters and the signal amplitudes should be adjusted to meet the
conditions of translational symmetry, often approximately, not exactly,
because of the dispersion slopes [17]. In particular, if one fiber span
has a positive-dispersion (+D) fiber followed by a negative-dispersion
(-D) fiber, then the counterpart span has to place the -D fiber before
the +D fiber, in order to achieve an approximate translational symmetry
between the two fiber spans. Even though the +D and -D fibers are usually
made of similar materials with similar nonlinear susceptibilities, their
guided-wave nonlinear coefficients measured in W-1km-1 could be
quite different due to the wide variation of modal sizes. Unless the
ratio of nonlinear coefficients matches the ratio of dispersions, the
signal powers in two conjugate fibers may have to differ by several dB as
required by the scaling rule of equation (33) for scaled translational
symmetry.

[0096] Should it be desired to have a similar level of signal powers into
the nonlinearity-compensating +D and -D fibers, one may adjust the signal
powers in the +D and -D fibers only slightly, such that one span of a
type with stronger nonlinearity generates an amount of nonlinearity that
is equivalent to an integral multiple of the amount of nonlinearity
generated in one span of another type with weaker nonlinearity. If each
span with weaker nonlinearity is dispersion-compensated to have
approximately zero accumulated dispersion, then each of several such
spans in cascade may indeed induce approximately the same nonlinear
response. And for a reasonably small number of such cascaded spans with
weaker nonlinearity, the overall nonlinear response may still be well
approximated by a combined first-order perturbation, which is just the
sum of the first-order perturbations of individual spans. Then one may
take advantage of the additivity of the first-order perturbations and
have one span of the type with stronger nonlinearity to compensate
several spans of the other type with weaker nonlinearity. This method may
be called "one-for-many" (in terms of fiber spans) nonlinearity
compensation. More generally, it is possible to have several spans of the
type with weaker nonlinearity generating different amounts of
nonlinearity, still their combined nonlinearity may be compensated by one
span of the type with stronger nonlinearity, so long as all
nonlinearities remain perturbative and the first-order perturbation of
the span with stronger nonlinearity is equivalent to the sum of the
first-order perturbations of the spans with weaker nonlinearity.

[0097] When two fiber spans are translationally symmetric about an optical
phase conjugator, one span is called the translational conjugate to the
other about the OPC. As argued above, OPC is able to equalize dispersion
terms of even orders. So the two parts of a transmission line with OPC in
the middle should have the same amount of b2 and b4 but exactly
opposite b3, or both have b3=0, where the b-parameters are
defined as in equation (13). In a more restrictive implementation, each
fiber span consists of +D and -D fibers with the total dispersion slope
compensated to zero. The +D and -D fibers in each span need not to match
their dispersions and slopes simultaneously. It is sufficient to fully
compensate b3, while leaving residual even-order terms b2 and
b4. Two conjugate spans would be configured as +D followed by -D
fibers and -D followed by +D fibers respectively. The two conjugate spans
may not be exactly the same in length, and they may have different
integrated dispersion terms of the even orders. The two types of fiber
spans may be mixed and alternated on each side of the OPC, so that the
two sides have the same total b2 and b4. Transmission lines
with such dispersion map are convenient to plan and manage. However, it
is worth noting that the present method of simultaneous compensation of
dispersion and nonlinearity applies to other dispersion maps as well,
where the period of dispersion compensation may be either shorter [38] or
longer [39] than the amplifier spacing, or the fiber spans may vary
widely in length and configuration. Regardless of the dispersion map,
wide-band dispersion compensation could be achieved in a transmission
line with middle-span OPC so long as the dispersion terms of the two
sides of OPC satisfy equation (14), and pairs of conjugate fiber spans
could have their nonlinearities cancelled up to the first-order
perturbation as long as the scaling rules of equations (29-32) and (33)
are well observed.

[0098] As a result of power loss, the nonlinear response of a long piece
of fiber becomes insensitive to the actual fiber length so long as it far
exceeds the effective length [2] defined as Leff=1/α, where
α is the loss coefficient in units of km-1 (instead of dB/km).
So fiber spans consisting of the same types of fibers but with different
lengths could contribute the same amount of nonlinearity if the input
powers are the same. That all fiber spans contribute the same
nonlinearity makes it possible for various spans with different lengths
to compensate each other's nonlinear effects. It is straightforward to
extend the same argument to fiber spans with scaled parameters and signal
powers. The conclusion is that scaled fiber spans could induce
approximately the same amount of nonlinear distortion to optical signals,
which is insensitive to the varying span lengths, provided that the
length of each fiber span is much longer than its own effective length
defined by the inverse of the loss coefficient. The main advantage is
that the fiber spans may be arbitrarily paired for nonlinearity
compensation regardless of their actual lengths. This is good news to
terrestrial and festoon systems, where the span-distance between
repeaters may vary according to the geographical conditions. When the
dispersion of each fiber span is not fully compensated, it is desirable
to fine-tune (slightly elongate or shorten) the lengths of transmission
fibers or DCFs such that all spans have the same amount of residual
dispersion. As a consequence, fiber spans of different lengths and
possibly consisting of different types of fibers become truly equivalent
in two all-important aspects of signal propagation: nonlinearity and
accumulated dispersion. Certainly, if the above-mentioned method of
"one-for-many" nonlinearity compensation is employed, the residual
dispersion of the highly nonlinear span should also be multiplied by the
same integer factor. Last but not least, when scaling fiber parameters
and signal amplitudes to have two fiber spans inducing the same or
compensating nonlinear effects, it is only necessary to make sure that
the scaling rules of equations (29-32) and (33) are fulfilled in portions
of fibers experiencing high levels of signal power. Elsewhere, the
scaling rules may be loosened or neglected when the signal power is low.

[0099] Despite the translational symmetry between the constituent fibers
of two conjugate spans, it is advantageous to order many conjugate spans
in a mirror-symmetric manner about the OPC, especially when all the spans
are not identical. The local nonlinearity within each span is usually
weak such that the nonlinear perturbations of higher orders than the
first may be neglected, even though a strong nonlinearity may be
accumulated through many fiber spans. Within the applicability of
first-order perturbation for approximating the nonlinearity of each fiber
span, it may be argued using mathematical induction that the nonlinearity
of multiple spans in cascade is also compensated up to the first-order
perturbation, because of the mirror-symmetric arrangement of fiber spans
about the OPC. The spans may be labelled from left to right by -N, . . .
, -2, -1, 1, 2, . . . , N, with OPC located between span -1 and span 1.
And one may denote by z0 and z0' the beginning and end
positions of the section of OPC, while labelling the beginning and end
points of span n by zn and zn', where zn'=zn+1,
.A-inverted.nε[-N, N-1]. There may be three variations for a
mirror-symmetric configuration of pairs of fiber spans in scaled
translational symmetry, depending upon whether the dispersion in each
span is compensated to zero, and if not, how the dispersion is managed.
In the first case, all spans are compensated to zero dispersion, as shown
in FIG. 14 for the case of N=3. It is required that,
.A-inverted.nε[1, N], spans -n and n should be conjugate, that is
translationally symmetric, to each other. The first-order nonlinear
perturbations of spans 1 and -1 cancel each other due to the
translational symmetry and the OPC, so the optical path from z-1 to
z1' is equivalent to an ideal linear transmission line with OPC in
the middle, if higher-order nonlinear perturbations are neglected. It
follows that the signal input to span 2 at z2 is approximately the
complex conjugate of that input to span -2 at z-2, apart from the
nonlinear perturbation due to span -2. So the translational symmetry
between spans 2 and -2 about the OPC annihilates their nonlinearities up
to the first-order perturbation. Using mathematical induction, assuming
that the optical path from z.sub.-n to zn', 1<n<N, is
equivalent to an ideal linear transmission line with OPC in the middle,
then spans n+1 and -n-1 see input signals at zn+1 and z.sub.-n-1
that are approximately complex conjugate to each other, so their
first-order nonlinear effects cancel each other out due to the
translational symmetry and OPC. The optical path from z.sub.-n-1 to
zn+1' is linearized and equivalent to an ideal linear transmission
line with OPC in the middle. This inductive argument applies as long as
the accumulation of nonlinear perturbations of higher-orders than the
first is still negligible and the nonlinear mixing of amplifier noise
into signal hasn't grown significantly.

[0100] In the second case, the fiber spans may have non-zero residual
dispersion, as shown in FIG. 15 for the case of N=3. It is required that,
.A-inverted.nε[1, N], spans -n and n should be in a translational
symmetry approximately, while the residual dispersion of span n-1 should
be approximately the same as span -n, .A-inverted.nε[2, N]. Pre-
and post-dispersion compensators are employed to equalize the residual
dispersion. The pre-dispersion may set the total dispersion to zero
immediately before OPC, and a dispersion conditioner at the site of OPC
ensures that the signal input to span 1 is approximately the complex
conjugate of that input to span -1, apart from the nonlinear perturbation
due to span -1. FIG. 15 shows a dispersion conditioner placed immediately
after OPC, with the amount of dispersion equal to the residual dispersion
in span -1. The three thicker line segments in the dispersion map
represent the effects of the pre- and post-dispersion compensators as
well as the dispersion conditioner. So the transmission line has been
designed such that the accumulated dispersions from z.sub.-n to zn,
nε[1, N], are fully compensated by virtue of OPC, and for each
nε[1, N], the fiber span from z.sub.-n to zn is
translationally symmetric to the fiber span from zn to zn',
namely, the parameters of the two fiber spans satisfy the scaling rules
of equations (29-32), at least approximately. Leaving aside the fiber
nonlinearity, such dispersion map ensures that the optical signals at
z.sub.-n and zn are complex conjugate to each other, then the signal
amplitudes may be properly scaled such that equation (33) is also
satisfied. As a result, all conditions are fulfilled for the fiber spans
from z.sub.-n to z.sub.-n' and from zn to zn' to compensate
their fiber nonlinearities up to the first-order perturbation, for each
nε[1, N]. The first-order nonlinear perturbations of spans 1 and
-1 cancel each other due to the translational symmetry and OPC, so the
optical path from z-1 to z1' is equivalent to an ideal linear
transmission line with OPC in the middle and some accumulated dispersion
at z1' due to span 1. Since this amount of dispersion is equal to
that of span -2, the signal input to span 2 at z2 is approximately
the complex conjugate of that input to span -2 at z-2, apart from
the nonlinear perturbation due to span -2. So the translational symmetry
between spans 2 and -2 about the OPC annihilates their nonlinearities up
to the first-order perturbation. Using mathematical induction, assuming
that the optical path from z.sub.-n to zn', 1<n<N, is
equivalent to an ideal linear transmission line with OPC in the middle
and accumulated dispersion at the right end due to span n, which is the
same amount of residual dispersion as of span -n-1, then spans n+1 and
-n-1 see input signals at zn+1 and z.sub.-n-1 that are approximately
complex conjugate to each other, so their first-order nonlinear effects
cancel each other out due to the translational symmetry and OPC. The
optical path from z.sub.-n-1 to zn+1' is linearized and equivalent
to an ideal linear transmission line with OPC in the middle and the
dispersion of span n+1 at the right end. In the third case, the fiber
spans still have non-zero residual dispersion, but there is no dispersion
conditioner placed immediately before or after OPC to compensate the
residual dispersion of span -1. Instead, span 1 may play the role of the
dispersion conditioner, and .A-inverted.nε[1, N], spans n and -n
need to have the same amount of residual dispersion, while spans n and
-n+1, .A-inverted.nε[2, N], should be in a scaled translational
symmetry approximately to have their nonlinearities compensated up to the
first-order perturbation. This is in contrast to the requirement of the
second case. The configuration is shown in FIG. 16 for the case of N=3,
where the two thicker line segments in the dispersion map represent the
effects of the pre- and post-dispersion compensators. It may be shown
using the same inductive argument that the transmission line is largely
linearized, except that the nonlinear effects of spans 1 and -N, if any,
are left uncompensated.

[0101] DCFs are widely used in modern fiber-optic transmission systems. A
DCF may be coiled into a compact module at the amplifier site, or cabled
as part of the transmission line. The performance of both types of DCFs
has been greatly improved recently. There are now low-loss DCFs capable
of (approximately) slope-matched dispersion compensation for various
transmission fibers with different ratios of dispersion to
dispersion-slope [3, 4], although there are always residual second-order
and fourth-order dispersions after the slope is equalized [5, 6, 7]. For
SMFs, namely standard single-mode fibers, the ratio of dispersion
(D≈16 ps/nm/km @1550 nm) to dispersion slope (S≈0.055
ps/nm2/km @1550 nm) is large, so that the relative change of
dispersion is small across the signal band (≈40 nm in the
C-band). The so-called reverse dispersion fibers (RDFs) are designed to
compensate simultaneously the dispersion and dispersion slope of the
SMFs. An RDF is not an ideal translational conjugate to an SMF, because
their dispersion slopes do not obey the scaling rule of equation (30).
However, their dispersions satisfy the corresponding scaling rule of
equation (30) approximately, with only small deviations across the entire
signal band (C or L). Therefore, a span having an SMF followed by an RDF
on the left side of OPC may be brought into a translational symmetry,
approximately, to a span having an RDF followed by an SMF on the right
side of OPC, and vice versa. The two types of spans may be denoted by
SMF+RDF and RDF+SMF respectively. The indication is that OPC may be
installed in the middle of conventional transmission lines with no or
minimal modifications to achieve simultaneous wide-band dispersion
compensation and nonlinearity suppression. The only requirements are that
the signal power levels should be properly set in the fiber spans, and
the SMFs/RDFs should be suitably arranged, to meet the scaling rules of
equations (29-32) and (33) approximately for the translational symmetry
between each pair of conjugate fiber spans, and to order the conjugate
pairs of spans mirror-symmetrically about the OPC. It is noted that a
recent paper [40] has independently proposed the combination of
slope-matching DCF and OPC to suppress simultaneously the third-order
dispersion and sideband instability due to fiber nonlinearity. However,
the work [40] was limited to a single-channel system, considered only the
suppression of sideband instability as an intra-channel nonlinear effect,
and did not recognize the importance of scaling the nonlinearity
(especially the signal power) in different fibers. By contrast, our
method applies to wide-band WDM systems as well and is capable of
suppressing both intra- and inter-channel nonlinear interactions, being
them Kerr- or Raman-originated. Most importantly, we emphasize the
importance of the scaling rules of equations (29-32) and (33) for optimal
nonlinearity compensation.

[0102] There have been RDFs [5, 7] with loss and dispersion coefficients
comparable to those of SMFs, namely,
αRDF≈αSMF≈0.2 dB/km,
DRDF≈-DSMF≈-16 ps/nm/km at about 1550 nm.
However, the effective modal area of RDFs is usually small, for example,
about 30 μm2, which is far less than the about 80 μm2
effective modal area of SMFs. Because the fiber nonlinear coefficients
are inversely proportional to the effective modal area, the example RDF
has Kerr and Raman nonlinear coefficients that are approximately
80/30≈2.7 times of those of SMFs, namely,
γRDF≈2.7γSMF,
gRDF(•)≈2.7 gSMF(•). If the same level of
signal power should be injected into the SMF+RDF and RDF+SMF spans, then
the amount of nonlinearity generated by an RDF+SMF span would be about
2.7 times of that generated by an SMF+RDF span. Nevertheless, one may
raise the input power to the RDF+SMF spans (relative to the input power
to the SMF+RDF spans) by only 0.3/2.7≈10% to have one RDF+SMF
span generates the equivalent amount of nonlinearity of, hence compensate
the nonlinear effects of, three SMF+RDF spans. Alternatively, one may
lower the input power to the RDF+SMF spans (relative to the input power
to the SMF+RDF spans) by just 0.7/2.7≈26% to have one RDF+SMF
span generates the equivalent amount of nonlinearity of, hence compensate
the nonlinear effects of, two SMF+RDF spans. These are practical examples
of the above-mentioned method of "one-for-many" nonlinear compensations.

[0103] Several non-zero dispersion-shifted fibers (NZDSFs) have also been
developed for long-distance high-capacity transmissions. These fibers
have reduced but non-zero dispersions across the operating band (C or L).
Depending upon the sign of the dispersion (D in units of ps/nm/km), there
are positive NZDSFs (+NZDSFs) and negative NZDSFs (-NZDSFs), but their
dispersion-slopes are always positive. It becomes possible to bring a
+NZDSF and a -NZDSF into a nearly perfect translational symmetry [41],
because their oppositely signed dispersions and positively signed
dispersion-slopes meet the exact requirements of the scaling rules of
equation (30). The dispersion-slope of the NZDSFs may be compensated by
negative-slope DCFs. The DCFs do not have to (could not indeed)
compensate the dispersion and dispersion-slope simultaneously for both
the positive and negative NZDSFs. It is sufficient to equalize the
accumulated dispersion-slope to zero on each side of the OPC, then the
two sides may cancel their accumulated non-zero dispersions of the second
and the fourth orders through OPC. To form a nonlinearity-compensating
translational symmetry between a +NZDSF span and a -NZDSF span, the
accumulated dispersion should be properly managed to ensure that the
input signals to the +NZDSF and -NZDSF fibers are complex conjugate to
each other, which is a necessary condition for nonlinearity cancellation.
As long as these requirements are satisfied, there is really no limit as
to how much residual (second-order) dispersion may be accumulated in each
fiber span as well as on each side of the OPC. It may be difficult to
find a fiber translationally symmetric to the slope-compensating DCF,
because of its high negative dispersion-slope. However, we note that it
is only necessary to have a scaled translational symmetry formed between
portions of fibers carrying high signal power, elsewhere, such as in the
slope-compensating DCFs, the scaling rules may be neglected when the
signal power is low and the nonlinearity is insignificant. If the
slope-compensating DCFs are cabled, they may be placed near the end of
fiber spans where the signal power is low. Or if the DCFs are coiled into
modules and co-located with the amplifiers, the signal power inside may
be controlled at a low level to avoid nonlinearity. To minimize the
noise-figure penalty in such DCF modules, the DCF may be distributively
Raman pumped [18, 34, 35], or earth-element doped and distributively
pumped [19], or divided into multiple segments and power-repeated by a
multi-stage EDFA. The conclusion is that the method of OPC-based
simultaneous compensation of dispersion and nonlinearity is perfectly
suitable for transmission systems employing NZDSFs, and highly effective
nonlinearity suppression may be expected in such systems due to the
nearly perfect translational symmetry between the +NZDSFs and -NZDSFs.
Finally, in the limit of vanishing (second-order) dispersion at the
center of the signal band, the +NZDSF and -NZDSF converge to the same
dispersion-shifted fiber (DSF), which is translationally symmetric to
itself. Two identical DSF spans on the two sides of OPC are in perfect
translational symmetry to cancel their nonlinearity up to the first-order
perturbation. Again the dispersion-slope may be equalized by a DCF with
negative dispersion-slope, and the residual second-order dispersion may
be arbitrarily valued. Suppressing fiber nonlinearity happens to be
highly desired in DSF-based transmission lines, as DSFs are arguably the
most susceptible to nonlinear impairments [2].

Simulation Results and Discussions

[0104] To verify the proposed method of simultaneous compensation of
dispersion and nonlinearity, we have carried out a series of numerical
simulations using a commercial transmission simulator
(VPItransmissionMaker®, Virtual Photonics Inc.). Reference [17] has
presented an example of SMFs and DCF modules with nearly perfect match of
dispersion and slope. Here we consider a practical setup of SMFs and
cabled DCFs with residual dispersion, as shown in FIG. 17. One type of
span consists of 50 km SMF followed by 50 km DCF. The SMF has loss
coefficient α=0.2 dB/km, effective mode area Aeff=80
μm2, and dispersion parameters β2=-20.5 ps2/km,
β3=0.12 ps3/km at 193.1 THz. The corresponding dispersion
D=16 ps/nm/km and slope S=0.055 ps/nm2/km. The DCF mimics a
commercial RDF product [7], namely a reverse dispersion fiber, with
parameters (α', Aeff', β2', β3')=(0.2, 30,
18, -0.12), in the same units as for the SMF. The Kerr nonlinear index of
silica n2=2.6×10-20 m2/W. Practical DCFs often have
a loss coefficient that is slightly higher than the SMFs, so the optimal
design of the DCFs would have a dispersion |DDCF| slightly higher
than |DSMF| proportionally according to the scaling rules of
equations (29) and (30). The conjugate span has 40 km DCF followed by SMF
of the same length. Due to the smaller modal area, a lower power is
injected into the DCF to generate the compensating nonlinearity, in
accordance with the scaling rule for signal amplitudes in equation (33).
The shortened span length is to balance the noise figure between the two
types of spans. The two span types are also intermixed on each side of
the OPC to balance the residual dispersions. Alternatively, all fiber
spans may be the same in length, but the signal power injected to the
DCF+SMF spans should be 3/8 of that injected to the SMF+DCF spans, and
the DCF+SMF spans would add more noise to the optical signal than the
SMF+DCF spans. It is noted that the scaling rules are not obeyed at all
in the second part of each span, that is, in the DCFs of SMF+DCF spans
and in the SMFs of DCF+SMF spans. Fortunately, the second part of each
span experiences low signal power, in which the nonlinear effect is
negligible. Back to the setup of FIG. 17, where all EDFAs have the same
noise figure of 5 dB, each fiber loop recirculates five times, that gives
1000 km worth of fiber transmission on each side of the OPC. The inputs
are four 40 Gb/s WDM channels, return-to-zero (RZ) modulated with peak
power 20 mW, channel spacing 200 GHz. Each RZ pulse generator consists of
a continuous-wave laser followed by a zero-chirp modulator, which is
over-driven to produce a pulse train with the amplitude proportional to
cos(π/2 sin πωt), where ω is the bit rate. Therefore
the duty cycle of the pulses is 33%, if defined as the ratio of pulse
full-width-half-maximum to the time interval between adjacent bits. The
optical multiplexer and demultiplexer consist of Bessel filters of the
7th order with 3 dB bandwidth 80 GHz. The input data are simulated by
pseudo random binary sequences of order 7, and the simulation time window
covers 256 bits. The photo-detector is with responsivity 1.0 A/W and
thermal noise 10.0 pA/ {square root over (Hz)}. The electrical filter is
3rd order Bessel with 3 dB bandwidth 28 GHz. FIG. 18 shows the received
eye diagrams of the second channel out of the demultiplexer (DEMUX). The
top-right diagram shows the effect of nonlinearity compensation. For
comparison, the result of a fictitious transmission where no fiber has
any nonlinear effect is shown on the top-left of FIG. 18. To confirm that
the suppression of nonlinearity is indeed due to the translational
symmetry of conjugate spans about the OPC, the two diagrams at the bottom
of FIG. 18 show simulation results of altered configurations: one setup
has the same length of 50+50 km for and the same input power level to
both the SMF+DCF and the DCF+SMF spans, the other has on both sides of
OPC identical 100-km SMF+DCF spans carrying the same signal power. Both
altered setups suffer from severe nonlinear impairments.

[0105] For an example of "one-for-many" nonlinearity compensation, we have
simulated a trans-mission system using SMF+RDF and RDF+SMF spans, as
shown in FIG. 19. The system has an optical phase conjugator in the
middle, and on each side of OPC there is a loop recirculating four times.
Each loop consists of three SMF+RDF spans each consisting of 40 km SMF+40
km RDF+16 dB EDFA, and one RDF+SMF span consisting of 40 km RDF+40 km
SMF+16 dB EDFA. The SMF has loss α=0.2 dB/km, dispersion D=16
ps/nm/km, dispersion-slope S=0.055 ps/nm2/km, effective modal area
Aeff=80 μm2, and the RDF has α'=0.2 dB/km, D'=-16
ps/nm/km, S'=-0.055 ps/nm2/km, Aeff'=30 μm2, the EDFA
has noise figure 4 dB. The inputs are four 40 Gb/s channels, RZ modulated
with peak power 10 mW and duty cycle 33%. The channel spacing is 200 GHz.
The optical MUX and DEMUX consist of Bessel filters of the 7th order with
3 dB bandwidth 100 GHz. The system is configured such that each RDF+SMF
span on the left side corresponds to and compensates the nonlinear
effects of three SMF+RDF spans on the right side, and each RDF+SMF span
on the right side corresponds to and compensates the nonlinear effects of
three SMF+RDF spans on the left side. We have tried both a case with all
spans being injected exactly the same amount of signal power and a case
with the RDF+SMF spans being fed with 10% more power comparing to the
SMF+RDF spans. No observable difference is found in the transmission
performance, which indicates robustness of the system design against
reasonable parameter deviations. FIG. 20 shows a typical eye diagram of
the received optical signals at the end of transmission, which
demonstrates excellent signal quality after 2560 km transmissions. We
have also simulated two comparative systems to see how effective is the
method of "on-for-many" nonlinearity compensation with scaled
translational symmetry, one of which as shown in FIG. 21 has all
identical SMF+RDF spans on both sides of OPC, the other of which as shown
in FIG. 22 has no OPC in the middle. Everything else remains the same.
Lacking a scaled translational symmetry, both comparative systems are
seriously impaired by fiber nonlinearities, as shown by the typical eye
diagrams of received optical signals in FIG. 23.

[0106] For an example system using NZDSFs, we simulated a transmission
line consisting of twenty 100-km fiber spans with OPC in the middle, as
shown in FIG. 24, where each side of the OPC has a fiber loop circulated
five times. In each circulation, the optical signals go through 100 km
-NZDSF transmission followed by a two-stage EDFA with 10 km DCF in the
middle, then 100 km+NZDSF transmission followed by the same two-stage
EDFA and DCF. The +NZDSF has loss coefficient α=0.2 dB/km,
dispersion D=+4 ps/nm/km and slope S=0.11 ps/nm2/km at 193.1 THz.
The effective mode area is Aeff=70 μm2. The -NZDSF differs
only by D=-4 ps/nm/km. The Kerr nonlinear index of silica
n2=2.6×10-20 m2/W. The two-stage EDFA has 11+15=26
dB gain in total to repeat the signal power. The noise figure of each
stage is 5 dB. The DCF has α=0.6 dB/km, D=-40 ps/nm/km, S=-1.1
ps/nm2/km, Aeff=25 μm2, but nonlinearity neglected. The
transmitting and receiving ends are the same as in the above SMF/DCF
transmission. Input to the system are the same four-channel WDM signals,
and the peak power of the 40 Gb/s RZ pulses is also the same 20 mW. With
their nonlinear effects neglected, the DCFs do not participate directly
in nonlinearity compensation. Nevertheless, their compensation of the
dispersion-slope of the NZDSFs enables the OPC to effectively compensate
the dispersion over a wide frequency band, and helps to condition the
optical signals such that the inputs to two conjugate NZDSFs are mutually
complex conjugate. Note that the +NZDSF and -NZDSF spans are alternated
on each side of the OPC to balance the accumulated dispersion between the
two sides. Also note that the first -NZDSF span on the right side of OPC
is designed to compensate the nonlinearity of the last +NZDSF span on the
left side, and the second span on the right (+NZDSF) is to compensate the
second last span (-NZDSF) on the left, so on and so forth. It is
important for the +NZDSF spans to be well dispersion-compensated, so to
ensure that the input signals to the two conjugate spans of a
translationally symmetric pair are complex conjugate to each other, which
is a necessary condition for nonlinearity cancellation. However, there is
no limit as to how much residual dispersion may be in the -NZDSF spans.
Alternatively, each fiber span may be a concatenation of + and -NZDSFs.
One type of span may have a +NZDSF followed by a -NZDSF, then the
conjugate span would consist of the same fibers reversely ordered.
Consequently, all spans may use the same DCF for slope compensation, and
all accumulate the same dispersions of even orders. FIG. 25 shows the
received eye diagrams of the second channel out of the DEMUX. The top row
shows the results of nonlinear transmission and the comparing fictitious
transmission without fiber nonlinearity through the setup of FIG. 24. The
effectiveness of nonlinear compensation is remarkable. By contrast, the
bottom row of FIG. 25 shows severe degradations in the transmission
performance, when all -NZDSFs are replaced by +NZDSFs, so that the
transmission line consists of identical +NZDSF spans with DCFs
compensating both the dispersion and the dispersion-slope. The highly
effective nonlinearity compensation is expected as a result of the nearly
perfect translational symmetry between the +NZDSF and -NZDSF spans.
Furthermore, a nonlinearity-suppressed transmission line should manifest
behaviors of a linear system to some extent. Typical linear behaviors
include scalability and cascadability. Namely, using the same fiber spans
and simply by raising the signal power, it is possible to further the
transmission distance by increasing the number of fiber spans
before/after the OPC (scaling up), or by cascading several
OPC-compensated transmission lines all-optically (without optical to
electrical and electrical to optical signal conversions in the middle).
Both the scalability and the cascadability are confirmed via numerical
simulations, as shown in FIG. 26, where one eye diagram is for a system
with the number of spans doubled to 40 in total, and the other diagram is
obtained when cascading two identical 20-span transmission lines of FIG.
24. The eye diagrams are still of the second channel out of the DEMUX.

[0107] To test the effectiveness of nonlinear compensation for DSFs, we
evaluated numerically a transmission line consisting of twenty 50-km DSF
spans with OPC in the middle, as shown in FIG. 27. Each span has 50 km
DSF and at the end a two-stage EDFA with 5 km DCF in the middle. The DSF
has loss α=0.2 dB/km, D=0 ps/nm/km and S=0.08 ps/nm2/km at the
center frequency 193.1 THz, Aeff=50 μm2. The Kerr nonlinear
index of silica is again n2=2.6×10-20 m2/W. The
two-stage EDFA has 6+7=13 dB gain in total to repeat the signal power,
and the noise figure of each stage is 5 dB. The DCF has α=0.6
dB/km, D=-100 ps/nm/km, S=-0.8 ps/nm2/km, Aeff=25 μm2,
but nonlinearity neglected. The transmitting and receiving ends are still
the same as in the above SMF/DCF transmission. However, the four channels
of 40 Gb/s RZ pulses are transmitted at (-350, -150, +50, +250) GHz off
the center frequency, and they are received at (-250, -50, +150, +350)
GHz off the center frequency. Note that the channels are assigned
asymmetrically about the center frequency to avoid phase-matched
four-wave mixing (FWM) [2]. The channels may also be unequally spaced to
further reduce the FWM penalty [42, 43]. But assigning channels with
unequal spacing increases the network complexity and may not provide
sufficient suppression by itself to the FWM and other nonlinear effects.
In particular, it is ineffective to suppress the effect of cross-phase
modulation (XPM). Nevertheless, when applicable, such legacy methods for
nonlinearity suppression may be combined with our method of OPC-based
nonlinearity compensation. The legacy methods may work to enhance the
effectiveness of our method, in the sense that they may render weaker
nonlinearity in each fiber span, so that the negligence of higher-order
nonlinear perturbations becomes a better approximation. Back to the
DSF-based transmission system of FIG. 27, when the power of the RZ pulses
is peaked at 2 mW, FIG. 28 shows the received eye diagrams of the second
channel out of the DEMUX. The top-left diagram is obtained when the fiber
nonlinearity is turned OFF, so the signal is only impaired by amplifier
noise. The top-right is the received eye diagram when the fiber
nonlinearity is turned ON. The increased penalty due to fiber
nonlinearity is visible but not too large. The eye diagrams at the bottom
of FIG. 28 are obtained when the dispersion of the DCFs changes to D=0
ps/nm/km while the slope remains, with or without OPC in the middle of
the link. The good transmission performance shown in the bottom-left
diagram verifies the insensitivity of our OPC-based method of
nonlinearity compensation to the amount of residual dispersion in each
fiber span, while the bad result on the bottom-right demonstrates the
indispensability of OPC.

Compensating Intra-Channel Nonlinear Effects Without OPC

[0108] When there is no optical phase conjugator available, two fiber
spans in a translational symmetry may still cancel out their
intra-channel nonlinear effects to a large extent, and a proper
arrangement of the pairs of translationally symmetric fiber spans could
significantly reduce intra-channel nonlinear effects in a long distance
transmission line. The intra-channel nonlinear effects, namely, nonlinear
interactions among optical pulses within the same wavelength channel, are
the dominating nonlinearities in systems with high modulation speeds of
40 Gb/s and above [39], where the nonlinear interactions among different
wavelength channels become less-limiting factors. As a result of short
pulse width and high data rate, optical pulses within one channel are
quickly dispersed and overlap significantly so to interact through the
Kerr effect. In the past a few years, intra-channel nonlinearities have
been extensively investigated by several research groups [44, 45, 46, 47,
48, 49, 50, 51, 52], and a method has been identified for suppressing the
intra-channel nonlinearity-induced jitters in pulse amplitude and timing,
using lossless or Raman-pumped transmission lines manifesting a mirror
symmetry [46, 52]. As mentioned before, the loss of pump power makes it
difficult to maintain a constant gain in a long transmission fiber.
Consequently, the significant deviation of signal power variation from a
desired mirror-symmetric profile degrades the result of intra-channel
nonlinear compensation using mirror symmetry [53]. Nevertheless, we have
found that two fiber spans in a scaled translational symmetry could
cancel out their intra-channel nonlinear effects to a large extent
without resorting to OPC, and a significant reduction of intra-channel
nonlinear effects may be achieved in a multi-span system with
translationally symmetric spans suitably arranged.

[0109] This time the translational symmetry requires that the
corresponding fiber segments have the same sign for the loss/gain
coefficients but opposite second- and higher-order dispersions, which are
naturally satisfied conditions in conventional fiber transmission
systems, where, for example, a transmission fiber may be paired with a
DCF as symmetric counterparts. The scaled translational symmetry further
requires that the fiber parameters should be scaled in proportion and the
signal amplitudes should be adjusted to satisfy,

.A-inverted.zε[0, L] and .A-inverted.tε(-∞,
+∞), where α(z), β2(z), β3(z), and
γ(z) denote the loss coefficient, second-order dispersion,
third-order dispersion, and Kerr nonlinear coefficient respectively for
one fiber stretching from z=0 to z=L>0, while the primed parameters
are for the other fiber stretching from z'=0 to z'=L/R, R>0 is the
scaling ratio, A(z, t) and A'(z', t) are the envelopes of optical
amplitude in the two fiber segments respectively, whose initial values at
z=0 and z'=0 respectively are required to be complex conjugate,

where κεR is an arbitrary phase. Even though the effect of
dispersion slope may be neglected within a single wavelength channel, the
inclusion of the β3-parameters in the scaling rules of equation
(34) ensures that good dispersion and nonlinearity compensation is
achieved for each wavelength channel across a wide optical band. When a
pair of such fiber segments in scaled translational symmetry are
cascaded, and the signal power levels are adjusted in accordance with
equation (34), it may be analytically proved that both the timing jitter
and the amplitude fluctuation due to intra-channel nonlinear interactions
among overlapping pulses are compensated up to the first-order
perturbation of fiber nonlinearity, namely, up to the linear terms of the
nonlinear coefficient. Since the dispersive and nonlinear transmission
response is invariant under the scaling of fiber parameters and signal
amplitudes as in equations (34) and (35) [1], it is without loss of
generality to consider two spans that are in translational symmetry with
the ratio R=1 and γ(z=0)=γ'(z'=0). The cascade of such two
spans would constitute a transmission line stretching from z=0 to z=2L,
with the fiber parameters satisfying,

.A-inverted.zε[0, L] and .A-inverted.tε(-∞,
+∞). The translational symmetry is illustrated in FIG. 29 with
plots of signal power and accumulated dispersion along the propagation
distance.

[0110] It is only necessary to consider the Kerr nonlinearity within one
wavelength channel, while the Raman effect may be neglected. The
amplitude envelope of a single channel may be represented by a sum of
optical pulses, namely, A(z, t)=Σkuk(z, t), where
uk(z, t) denotes the pulse in the kth bit slot and centered at time
t=kT, with kεZ and T>0 being the bit duration. The following
NLSE describes the propagation and nonlinear interactions among the
pulses [39],

where the right-hand side keeps only those nonlinear products that
satisfy the phase-matching condition. The nonlinear mixing terms with
either m=k or n=k contribute to self-phase modulation and intra-channel
XPM, while the rest with both m≠k and n≠k are responsible for
intra-channel FWM [39]. It is assumed that all pulses are initially
chirp-free or they can be made so by a dispersion compensator, and when
chirp-free the pulses uk (z=0, t), kεZ, should all be
real-valued. This includes the modulation scheme of binary phase-shift
keying, where the relative phases between adjacent pulses are either 0 or
π. It is only slightly more general to allow the pulses being modified
by arithmetically progressive phase shifts
φk=φ0+kΔφ, kεZ, with φ0,
Δφε[0, 2π, because equation (37) is invariant under
the multiplication of phase factors exp(iφk) to uk,
.A-inverted.kεZ. The linear dependence of φk on k is in
fact equivalent to a readjustment of the frequency and phase of the
optical carrier. The pulses may be RZ modulated, and non-return-to-zero
(NRZ) modulated as well, for an NRZ signal train is the same as a stream
of wide RZ pulses with 100% duty cycle.

[0111] Were there no nonlinearity in the fibers, the signal propagation
would by fully described by the dispersive transfer function,

which is calculated from F-1[H(z1, z2, ω)] up to a
constant phase factor. The impulse response defines a linear propagator
P(z1, z2) as in equation (9). In reality, the signal evolution
is complicated by the Kerr nonlinear effects. Nevertheless, the
nonlinearity within each fiber span may be sufficiently weak to justify
the application of the first-order perturbation theory:

.A-inverted.kεZ, where uk(z, t)≈vk(z, t) is the
zeroth-order approximation which neglects the fiber nonlinearity
completely, whereas the result of first-order perturbation uk(z,
t)≈vk(z, t)+vk'(z, t) accounts in addition for the
nonlinear products integrated over the fiber length. For the moment, it
is assumed that both fiber spans are fully dispersion- and
loss-compensated to simplify the mathematics. It then follows that b(0,
z+L)=-b(0, z),
∫0z+Lα(s)ds=∫0zα(s)ds,
γ(z+L)=γ(z), .A-inverted.zε[0, L], and vk(L,
t)=vk(2L, t)=uk(0, t), which is real-valued by assumption,
.A-inverted.kεZ. It further follows that h(0, z+L, t)=h*(0, z,
t), hence P(0, z+L)=P*(0, z) and P(z+L, 2L)=P*(z, 2L),
.A-inverted.zε[0, L]. Consequently, the pulses at z and z+L are
complex conjugate, namely, vk(z+L, t)=vk*(z, t),
.A-inverted.kεZ, .A-inverted.zε[0, L]. A typical term of
nonlinear mixing,

is therefore real-valued. It follows immediately that the first-order
nonlinear perturbation vk'(2L, t) is purely imaginary-valued, which
is in quadrature phase with respect to the zeroth-order approximation
vk(2L, t)=vk(0, t), .A-inverted.kεZ. When the span
dispersion is not fully compensated, namely, b2(0, L)≠0, the
input pulses to the first span at z=0 should be pre-chirped by an amount
of dispersion equal to -1/2b2(0, L), so that the input pulses to the
second span at z=L are pre-chirped by 1/2b2(0, L) as a consequence.
In other words, the input signals to the two spans should be oppositely
chirped. Under this condition, the equation vk(z+L, t)=vk'(z,
t), .A-inverted.zε[0, L], .A-inverted.kεZ is still valid,
so are the above argument and the conclusion that vk and vk'
are real- and imaginary-valued respectively when brought chirp-free.

[0112] Mathematically, that vk and vk' are in quadrature phase
implies |uk|2=|vk+vk'|2=|vk|2+|vk-
'|2, where |vk|2 is quadratic, or of second-order, in terms
of the Kerr nonlinear coefficient. This fact has significant implications
to the performance of a transmission line. Firstly, it avoids pulse
amplitude fluctuations due to the in-phase beating between signal pulses
and nonlinear products of intra-channel FWM, which could seriously
degrade the signal quality if not controlled [39, 45, 46, 51]. The
quadrature-phased nonlinear products due to intra-channel FWM lead to the
generation of "ghost" pulses in the "ZERO"-slots [44, 48, 49] and the
addition of noise power to the "ONE"-bits. As second-order nonlinear
perturbations, these effects are less detrimental. Secondly, it
eliminates pulse timing jitter due to intra-channel XPM up to the
first-order nonlinear perturbation. Using the moment method [45, 46], the
time of arrival for the center of the kth pulse may be calculated as,

which is clearly jitterless, .A-inverted.kεZ. In the calculation,
the |vk'|2 terms are simply neglected as they represent
second-order nonlinear perturbations.

[0113] Fiber spans for intra-channel nonlinearity compensation without OPC
may be similarly designed and arranged as those described in previous
sections when OPC is used. A trans-mission fiber, either SMF or NZDSF,
and its corresponding slope-matching DCF [3, 4] are a perfect pair for
compensating intra-channel nonlinearities, as their dispersions and
slopes of dispersion satisfy the scaling rules of equation (34)
perfectly, and the signal amplitudes may be easily adjusted to fulfil the
corresponding scaling rule. The so-called RDFs [7], as a special type of
DCFs, may be suitably cabled into the transmission line and contribute to
the transmission distance, since the absolute dispersion value and loss
coefficient of RDFs are both comparable to those of the transmission
fiber. Only the smaller modal area requires a lower level of signal power
for an RDF to compensate the nonlinearity of a transmission fiber.
Otherwise the "one-for-many" compensation scheme may be employed, where
the signal power may be slightly adjusted for an RDF to compensate the
nonlinearity of multiple transmission fibers. There is usually no power
repeater between the transmission fiber and the cabled RDF within one
span, so that the signal power decreases monotonically in each fiber
span, as shown in FIG. 29. Note that one fiber span has a transmission
fiber followed by an RDF, while the other span has an RDF followed by a
transmission fiber, in accordance with the scaling rules of equation (34)
for nonlinearity compensation. Alternatively, if distributive Raman
amplification, especially backward Raman pumping, is used to repeat the
signal power, then one span has the transmission fiber Raman pumped in
accordance with the RDF being Raman pumped in the other span. The signal
power variation in each span may no longer be monotonic, but the power
profiles in two compensating spans should still be similar and obey the
scaling rules of equation (34), especially in portions of fibers that
experience high signal power.

[0114] For DCFs having absolute dispersion values much higher than the
transmission fiber, it is suitable to coil the DCF into a lumped
dispersion-compensating module (DCM) and integrate the module with a
multi-stage optical amplifier at each repeater site. Two fiber spans in
scaled translational symmetry for intra-channel nonlinearity compensation
should have oppositely ordered transmission fibers and DCFs. As shown in
FIG. 30, one span has a piece of transmission fiber from A to B, in which
the signal power decreases exponentially, and an optical repeater at the
end, in which one stage of a multi-stage optical amplifier boosts the
signal power up to a suitable level and feeds the signal into a lumped
DCM, where the signal power also decreases exponentially along the length
of the DCF from B to C, finally the signal power is boosted by another
stage of the optical amplifier. The other span has the same transmission
fiber and the same DCM, with the signal power in the DCF from C to D
tracing the same decreasing curve. However, this span has the DCM placed
before the transmission fiber. Ironically, the efforts of improving the
so-called figure-of-merit [1, 4] by DCF vendors have already rendered the
loss coefficients of DCFs too low to comply with the scaling rules of
equation (34). To benefit from nonlinearity compensation enabled by
scaled translational symmetries, DCFs, at least parts of them carrying
high signal power, may be intentionally made more lossy during
manufacturing or by means of special packaging to introduce bending
losses. As illustrated in FIG. 30, the DCFs from B to C and from C to D
are arranged in scaled translational symmetry to the transmission fibers
from D to E and from A to B respectively, such that the transmission
fiber from A to B is compensated by the DCF from C to D, and the DCF from
B to C compensates the transmission fiber from D to E, for the most
detrimental effects of uttering in pulse amplitude and timing due to
intra-channel FWM and XPM. In practice, the DCMs from B to D and the
multistage optical amplifiers may be integrated into one signal repeater,
and the same super-span from A to E may be repeated many times to reach a
long-distance, with the resulting transmission line enjoying the
effective suppression of intra-channel nonlinear impairments. Again, in
case distributive Raman pumping in the transmission fibers is employed to
repeat the signal power, the DCFs may also be Raman pumped or
erbium-doped for distributive amplification to have similar (scaled)
power profiles as that in the transmission fibers for optimal
nonlinearity compensation.

[0115] It should be noted that in regions of fibers carrying lower optical
power, the scaling rules of fiber parameters in equation (34) may be
relaxed without sacrificing the performance of nonlinearity compensation,
both for systems using cabled DCFs into the transmission lines and for
systems using lumped DCMs at the repeater sites. Such relaxation may be
done for practical convenience, or to control the accumulated dispersion
in a span to a desired value, as well as to reduce the span loss so to
reduce the penalty due to optical noise. As an example and a potentially
important invention in its own right, a DCM compensating the dispersion
and nonlinearity of transmission fibers may be so packaged that the first
part of DCF experiencing a high level of signal power may have a higher
loss coefficient satisfying the scaling rule of equation (34), whereas
the second part of DCF may ignore the scaling rule and become less lossy
such that the signal power at the end of the DCM is not too low to be
significantly impaired by the amplifier noise. In fact, the low-loss part
of the DCM may even use optical filters other than DCFs, such as fiber
Bragg gratings and photonic integrated circuits. This method of packaging
DCMs achieves the capability of nonlinearity compensation and good noise
performance simultaneously. For instance, it takes 10 km DCF with D'=-80
ps/nm/km to compensate 100 km NZDSF with dispersion D=8 ps/nm/km and loss
α=0.2 dB/km. The first 4 km of the DCF may be made highly lossy by
a special treatment in manufacturing or packaging, with a loss
coefficient α'=2 dB/km to form a scaled translational symmetry with
respect to the first 40 km NZDSF for optimal nonlinearity compensation.
However, the remaining 6 km DCF may ignore the scaling rules and have a
much lower nominal loss α'=0.6 dB/km. The total loss is reduced by
8.4 dB as compared to a DCM that complies strictly with the scaling rules
throughout the length of the DCF. Another important parameter of DCFs is
the effective modal area, or more directly the nonlinear coefficient.
Traditional designs of DCFs have always strived to enlarge the modal area
so to reduce the nonlinear effects of DCFs. However, for DCFs used in our
method of nonlinearity compensation, there exists an optimal range of
modal area which should be neither too large nor too small. According to
the scaling rules of equation (34), a DCF with a large modal area may
require too much signal power to generate sufficient nonlinearity to
compensate the nonlinear effects of a transmission fiber, while optical
amplifiers may have difficulty to produce that much signal power. On the
other hand, when the effective modal area is too small, the scaling rules
of equation (34) dictate a reduced power level for the optical signal in
the DCF, which may be more seriously degraded by optical noise, such as
the amplified-spontaneous-emission noise from an amplifier at the end of
the DCF.

[0116] It is further noted that the nonlinear responses of fiber spans of
different lengths may be approximately the same so long as each of them
is much longer than the effective length Leff=1/α. This makes
nonlinearity compensation possible among spans with different lengths,
which are commonly seen in terrestrial and festoon systems, where the
span-distance between repeaters may vary according to the geographical
conditions. The dispersion of each fiber span may not be always fully
compensated, in which case it is desirable to fine-tune the fiber lengths
such that any pair of compensating spans have the same amount of residual
dispersion. The final note is that two compensating fiber spans are not
necessarily located immediately next to each other as drawn in FIGS. 29
and 30. Sometimes, it may be advantageous to order pairs of compensating
fiber spans in a mirror-symmetric manner similar to that discussed
previously, especially when all spans are not compensated to zero
dispersion. Indeed, it is convenient to have the two spans of any
compensating pair accumulating the same amount of total dispersion
including the sign. This would be achieved naturally if the two
compensating spans consist of exactly the same DCF and transmission fiber
of exactly the same lengths, with the only difference being the ordering
of the fibers. When a pair of compensating spans are not the same in span
distance, the length of either a DCF or a transmission fiber may be
fine-tuned, namely slightly elongated or shortened, to make sure that the
two spans have the same accumulated dispersion. If the spans of a
long-distance transmission line are labelled by -N, -N+1, . . . , -2, -1
and 1, 2, . . . , N-1, N from one end to the other, N>1, a
mirror-symmetric arrangement requires that spans -n and n, nε[1,
N] should be paired for nonlinearity compensation, that is, their fiber
parameters should satisfy the scaling rules of equation (34)
approximately and their accumulated dispersions should be the same. Note
that the scaling rules may only be fulfilled approximately if the two
spans have the same non-zero accumulated dispersion. Then pre- and
post-dispersion compensators may be employed at the two ends of the
transmission line to equalize the total dispersion and importantly, to
make sure that the accumulated dispersion from the transmitter to the
beginning of span -n is opposite to the accumulated dispersion from the
transmitter to the beginning of span n, for all nε[1, N], such
that the input signals to spans -n and n are complex conjugate, that is
oppositely chirped, as required for compensating intra-channel
nonlinearities. As an example, when all spans have the same ac cumulated
dispersion b2, the pre-dispersion compensator should provide

- ( N - 1 2 ) b 2 , ##EQU00021##

while the post-dispersion compensator should contribute

- ( N + 1 2 ) b 2 . ##EQU00022##

Or the amount of post-dispersion compensation may be slightly different
from

- ( N + 1 2 ) b 2 , ##EQU00023##

such that the overall dispersion of the transmission line is not zero but
within the tolerance of the transmitted pulses. More generally, if the
accumulated dispersions of spans -n and n are B.sub.-n and Bn
respectively, which satisfy the conditions B.sub.-n=Bn,
.A-inverted.nε[1, N], while Bm and Bn are not
necessarily the same if m≠n, then the pre- and post-dispersion
compensators may provide respectively

1 2 B 1 - n = 1 N B n and - 1 2
B 1 - n = 1 N B n ##EQU00024##

worth of dispersion, approximately up to the tolerance of the transmitted
pulses. It is worth pointing out that the single-channel nature of
intra-channel nonlinearity compensation permits the use of channelized
pre- and post-dispersion compensators. Namely, at each end of the
transmission line, apart from a common pre- or post-dispersion
compensator shared by all channels, each individual channel may have a
channelized dispersive element, or a short piece of fiber with the length
fine-tuned, to compensate the channel-dependence of dispersion if any.
Finally, it should be noted that a recent paper [54] proposes to
compensate the timing jitter due to intra-channel XPM in a transmission
fiber using the nonlinearity of a DCF, which is similar in spirit to our
method of intra-channel nonlinearity compensation using scaled
translational symmetry. However, the proposal in [54, 55] is limited to
the compensation of timing jitter of RZ pulses that are Gaussian-shaped,
whereas our method could compensate both the amplitude fluctuation and
timing jitter due to intra-channel nonlinear interactions of arbitrarily
shaped pulses, with the only condition for suppressing intra-channel FWM
that the signal pulses when chirp-free should be all real-valued upon a
suitable choice of frequency and phase for the optical carrier. More
importantly, the work presented in [54, 55] did not recognize the
significance of scaling the dispersion, loss and nonlinear coefficients
of the DCF with respect to the transmission fiber, which is a necessary
condition for optimal compensation of nonlinear effects. On the practical
side, the proposal in [54, 55] requires fiber Bragg grating dispersion
compensators, which are limited in operating bandwidth and may suffer
problems as thermal instability and group-delay ripples.

[0117] As usual, numerical simulations are carried out to support our
theoretical analysis and verify the effectiveness of our method of
suppressing intra-channel nonlinearity using scaled translational
symmetry. In one test system, as depicted in FIG. 31, the transmission
line consists of 6 pairs of compensating fiber spans totaling a
transmission distance of 1080 km. The first span in each pair has 50 km
SMF followed by 50 km RDF then an EDFA with gain 16 dB, the second span
has 40 km RDF followed by 40 km SMF then an EDFA with gain 20 dB. The
other test system consists of the same number of spans with the same span
lengths, which are constructed using the same fibers and EDFAs as the
first system except that the second span in each span-pair has the 40-km
SMF placed before the 40-km RDF, as shown in FIG. 32. The EDFA noise
figure is 4 dB. The SMF has loss α=0.2 dB/km, dispersion
D=16+δD ps/nm/km, and dispersion slope S=0.055 ps/nm2/km,
effective modal area Aeff=80 μm2, while the RDF has
α=0.2 dB/km, D=-16 ps/nm/km, S=-0.055 ps/nm2/km, and
Aeff=30 μm2. Fiber-based pre- and post-dispersion
compensators equalize 11/24 and 13/24 respectively of the total
dispersion accumulated in the transmission line. Both the SMF and the RDF
have the same nonlinear index of silica n2=2.6×10-20
m2/W. The transmitter has four 40 Gb/s WDM channels. The center
frequency is 193.1 THz, and the channel spacing is 200 GHz. All four
channels are co-polarized and RZ-modulated with 33% duty cycle and peak
power of 15 mW for the RZ pulses. The multiplexer (MUX) and DEMUX filters
are Bessel of the 7th order with 3 dB-bandwidth 80 GHz. The electrical
filter is third-order Bessel with 3 dB-bandwidth 28 GHz. The results of
four-channel WDM transmissions have been compared with that of
single-channel transmissions, with no clearly visible difference
observed, which indicates the dominance of intra-channel nonlinearity and
the negligibility of inter-channel nonlinear effects. Several trials with
various values for δD have been simulated for each test system. The
following figures present the eye diagrams of optical pulses after
wavelength DEMUX, in order to signify the nonlinear deformation (timing
and amplitude jitters) of optical pulses and the generation of
ghost-pulses. FIG. 33 shows the received optical pulses of δD=0 for
the two test systems, with the amplifier noise being turned off to
signify the nonlinear impairments (bottom diagram) and the effectiveness
of nonlinearity compensation (top diagram). Clearly shown is the
suppression of nonlinear impairments by using scaled translational
symmetry, and especially visible is the reduction of pulse timing jitter,
as seen from the thickness of the rising and falling edges as well as the
timing of pulse peaks. In both eye diagrams, there are optical pulses
with small but discernable amplitudes above the floor of zero signal
power, which could be attributed to ghost-pulse generation [44, 48, 49]
due to the uncompensated in-phase components of intra-channel FWM. When
the amplifier noise is turned back on, as shown in FIG. 34, the received
signals become slightly more noisy, but the suppression of nonlinear
distortions is still remarkable when there is scaled translational
symmetry. Then δD=0.2 ps/nm/km was set for the two test systems of
FIG. 31 and FIG. 32 respectively, in order to showcase that a
mirror-symmetric ordering of pairwise translationally symmetric fiber
spans is fairly tolerant to the residual dispersions in individual fiber
spans. In this setting, each fiber span has 10 or 8 ps/nm/km worth of
residual dispersion, and the accumulated dispersion totals 108 ps/nm/km
for the entire transmission line. Importantly, the pre- and
post-dispersion compensators are set to compensate 11/24 and 13/24
respectively of the total dispersion, ensuring at least approximately the
complex conjugateness between the input signals to each pair of spans in
scaled translational symmetry. The amplifier noise is also turned on. The
transmission results, as shown in FIG. 35, are very similar to that with
6D=0, which demonstrates the dispersion tolerance nicely. In a better
optimized design to tolerate higher dispersion mismatch |δD|,
either SMFs or RDFs may be slightly elongated or shortened in accordance
with the value of δD, such that the same residual dispersion is
accumulated in all spans. As an example, δD is set to 0.6 ps/nm/km
and each 40-km SMF is elongated by about 0.4 km, so that all spans have
the same residual dispersion of 30 ps/nm/km, and the whole transmission
line accumulates 360 ps/nm/km worth of dispersion. The pre- and
post-dispersion compensators equalize 360×11/24=165 and
360×13/24=195 ps/nm/km worth of dispersion respectively. The
amplifier noise is still on. The transmission results are shown in FIG.
36.

[0118] For an example of intra-channel nonlinear compensation using
"one-for-many" scaled translational symmetry, we have simulated an
optimized system using SMF+RDF and RDF+SMF spans as shown in FIG. 37 and
a comparative system using all SMF+RDF spans as shown in FIG. 38. Each
system is a cascade of two identical transmission lines. Each
transmission line has a loop recirculating twice, with each loop
consisting of four spans. In the optimized system, each loop consists of
three SMF+RDF spans each consisting of 40 km SMF+40 km RDF+16 dB EDFA,
and one RDF+SMF span consisting of 40 km RDF+40 km SMF+16 dB EDFA. Each
loop in the comparative system has all four identical SMF+RDF spans
consisting of 40 km SMF+40 km RDF+16 dB EDFA. The SMF has loss
α=0.2 dB/km, dispersion D=16 ps/nm/km, dispersion-slope S=0.055
ps/nm2/km, effective modal area Aeff=80 μm2, and the
RDF has α'=0.2 dB/km, D'=-16 ps/nm/km, S'=-0.055 ps/nm2/km,
Aeff'=30 μm2, the EDFA has noise FIG. 4 dB. The inputs are
four 40 Gb/s channels, RZ modulated with peak power 10 mW and duty cycle
33%. The channel spacing is 200 GHz. The optical MUX and DEMUX consist of
Bessel filters of the 7th order with 3 dB bandwidth 100 GHz. Note that
the optimized system is configured such that each RDF+SMF span
corresponds to and compensates the intra-channel nonlinear effects of
three SMF+RDF spans. Again we have tried both a case with all spans being
injected exactly the same amount of signal power and a case with the
RDF+SMF spans being fed with 10% more power comparing to the SMF+RDF
spans. No observable difference is found in the transmission performance,
which indicates robustness of the system design against reasonable
parameter deviations. The comparative system has no "one-for-many" scaled
translational symmetry. FIG. 39 shows typical eye diagrams of the
received optical signals at the mid-span of the transmissions, namely,
after the first transmission line of 640 km for each system. The eye
diagram of the optimized system shows significantly reduced amplitude and
timing jitters than the one of the comparative system, which demonstrates
the effect of intra-channel nonlinear compensation with the
"one-for-many" scaled translational symmetry. At the end of the 1280 km
transmissions, as shown in FIG. 40, the comparative system suffers from
significantly more signal degradations than the optimized system,
especially in terms of amplitude and timing jitters of the mark pulses.
However, it is noted that the optimized system is also penalized by the
accumulation of noise energy in the "originally empty" time slots.

[0119] In high-speed long-distance fiber-optic transmissions, a major
limitation is imposed by the intra-channel nonlinear effects, such as the
pulse amplitude and timing jitters due to intra-channel cross-phase
modulation (IXPM) and intra-channel four-wave mixing (IFWM) respectively
[39]. A method has been proposed to suppress the intra-channel
nonlinearities using Raman-pumped transmission lines manifesting a
lossless or mirror-symmetric map of signal power [46, 52]. However, the
loss of pump power makes it difficult to maintain a constant gain in a
long transmission fiber. Consequently, the significant deviation of
signal power profile from a desired mirror-symmetric map degrades the
result of intra-channel nonlinear compensation using mirror symmetry
[53]. The above has shown that transmission lines designed with
translational symmetries in power and dispersion maps could also
effectively compensate the IXPM and one aspect of IFWM, so to greatly
reduce the timing and amplitude jitters. There have also been recent
publications along the similar direction [54, 55]. In particular, our
mathematical formulation in the previous section provides a general and
unified theory for intra-channel nonlinearity compensation using
translational or mirror symmetry, and more importantly, it emphasizes the
necessity of scaling dispersion, loss coefficient, as well as the product
of nonlinear coefficient and signal power in fibers, for optimal
nonlinearity compensation. The one aspect of IFWM refers to amplitude
fluctuation in the "pulse-ON" slots due to coherent superpositions of
nonlinearly generated fields onto the original pulses. However, neither
the mirror nor the translational symmetry could hold back another aspect
of IFWM, namely, the generation of "ghost-pulses" into the "pulse-OFF"
slots where there are originally no optical pulses [44, 48, 56, 57]. The
growth of ghost-pulses will eventually limit the transmission distance.
Here we show that self-phase modulation (SPM) in the middle could make
the two parts of a long transmission line generating oppositely signed
ghost amplitudes, such that the ghost-pulses are annihilated or greatly
suppressed at the end.

[0120] The amplitude envelope of a single channel may be represented by a
sum of optical pulses, namely, A(z, t)=>Σkuk(z, t),
where uk(z, t) denotes the pulse in the kth time slot and centered
at time t=kT, with kεZ and T>0 being the duration of one
symbol. Again, the following nonlinear Schrodinger equation describes the
propagation and nonlinear interactions among the pulses [39],

where the right-hand side keeps only those nonlinear products that
satisfy the phase-matching condition. The nonlinear mixing terms with
either m=k or n=k contribute to SPM and IXPM, while the rest with both
m≠k and n≠k are responsible for IFWM [39]. For a pulse-OFF
time slot, for example the kth, the original pulse amplitude uk(0,
t)=0, however the Kerr nonlinearity will generate a ghost amplitude into
this slot. In the regime of weak nonlinearity where perturbation theory
applies, the ghost amplitude is approximated by a linear accumulation of
nonlinear products over the propagation distance,

[0121] Consider two transmission lines in cascade, one stretching from z=0
to z=L, the other from z=L to z=L+L'. Assuming dispersion is compensated
in each line such that optical pulses "return" approximately to their
original shapes at z=L and z=L+L'. Each line may consist of multiple
power-repeated and dispersion-equalized fiber spans which are suitably
arranged to form a scaled translational or mirror symmetry. Therefore,
both lines are effective for suppressing the timing and amplitude jitters
in the pulse-ON slots. However, they are not able to prevent the growth
of ghost amplitudes in the pulse-OFF slots. The two lines are not
necessarily the same, but assumed to generate approximately the same
ghost amplitudes, namely,

for all pulse-OFF slots labelled by k. So the ghost amplitude will
accumulate into uk(L+L', t)=2uk(L, t) at the end, as long as
the perturbation assumption still holds. If the trans-mission lines
become too long, the approximation of linear accumulation of ghost
amplitudes will eventually break down. The ghost amplitudes will actually
grow exponentially as a result of parametric amplification pumped by the
mark pulses. A method of ghost-pulse suppression may need to clean the
ghost amplitudes or start reversing their accumulation before they become
too strong.

[0122] Now consider introducing a self-phase modulator for each wavelength
channel in the middle of the two lines at z=L, and adjusting the signal
power such that the amount of nonlinear phase shift reaches π
approximately at the peak of an optical pulse. FIG. 41 shows such two
transmission lines with channelized SPM in the middle, where each
trans-mission line is scaled translationally symmetrically configured for
intra-channel nonlinearity compensation. After mid-span SPM, all
"originally ON" pulses acquire approximately a π phase shift, while
the ghost-pulses in the "originally OFF" time slots experience negligible
to small phase shifts due to their low power level. As a consequence, the
IFWM products generated in the second line from z=L to z=L+L' would
acquire a factor (-1)3=-1 with respect to when mid-span SPM is
absent. For a typical pulse-OFF slot labelled by k, the following
calculation gives the ghost amplitude generated from start to end through
the two transmission lines with SPM in the middle,

according to equation (47). Instead of adding up constructively, the
ghost amplitudes generated by the two lines interfere destructively to
cancel each other at the end z=L+L'. Good transmission performance may be
expected from the overall system, as a result of the suppression of
amplitude and timing jitters for originally ON pulses and the elimination
of ghost-pulses in the originally OFF time slots.

[0123] For implementations, the self-phase modulator may be based on the
fiber Kerr nonlinearity [25], cascaded χ.sup.(2) in LiNbO3
waveguides [58, 59], the index change induced by carrier density
variations in semiconductor optical amplifiers [60], or a combination of
a photodiode detecting the optical pulses and electro-optic phase
modulator driven by the generated electrical pulses [61, 62]. A
fiber-based self-phase modulator may be a better choice than others
because of its simplicity and capability of polarization-insensitive
operation. Furthermore, a suitable value of fiber dispersion may be
chosen such that optical pulses propagate in a soliton-like manner
through the nonlinear fiber, in order to reduce the pulse spectral
broadening due to SPM [25]. If SPM is not properly balanced by
dispersion, then only the peak of a pulse receives a 7 phase shift, the
rising and falling edges experience less and varying phase shifts, which
lead to frequency chirp and spectral broadening. Excessive spectral
broadening may cause crosstalk among wavelength channels and decrease the
spectral efficiency (rate of data transmission in bit/s over available
optical bandwidth in Hz) of transmission systems. A soliton, namely a
hyperbolic secant pulse, could propagate invariantly in a lossless fiber
given the condition -β2=γP0T02, where
β2 and γ are the dispersion and nonlinear coefficients of
the fiber, P0 and T0 are the peak power and width parameter of
the pulse [25]. For actual fibers with loss, strict soliton propagation
may not be possible, but the total fiber dispersion may be adjusted so to
minimize the frequency chirp of pulses at the end, or to control the
chirp at a desired level. An optical filter may also be employed after
SPM to limit the spectral width of pulses.

[0124] For numerical verifications, we have simulated and compared the
performance of three transmission lines, all of which use SMFs with loss
α=0.2 dB/km, dispersion D=16 ps/nm/km, effective modal area
Aeff=80 μm2, and RDFs, namely reverse dispersion fibers,
with loss α'=0.2 dB/km, dispersion D'=-16 ps/nm/km, effective modal
area Aeff'=30 μm2, as well as EDFAs with noise FIG. 4 dB.
The first setup is a conventional design consisting of 16 fiber spans,
each span has 45 km SMF, followed by 45 km RDF, and a 18 dB EDFA at the
end. The second setup is configured to form a scaled translational
symmetry, having 8 repetitions of (50 km SMF+50 km RDF+16 dB EDFA)+(40 km
RDF+40 km SMF+20 dB EDFA). Note that the EDFA gains are set in a way that
the signal powers into the 50-km SMF and the 40-km RDF are properly
scaled. The third system is the same as the second, except for
channelized SPM in the middle, using a high-power EDFA, an optical
DEMUX/MUX pair, and for each channel a 10-km nonlinear fiber with
effective modal area Aeff''=20 μm2, loss α''=0.3
dB/km, and dispersion D''≈3 ps/nm/km. The optical power is
boosted to 80 mW before entering each SPM fiber, and attenuated back to
the nominal level for transmissions after the self-phase modulator. All
fibers are made of silica glass with nonlinear index
n2=2.6×10-20 m2/W. Input to all three systems are
four 40 Gb/s channels, spaced by 200 GHz, co-polarized, and
return-to-zero modulated with 33% duty and peak power 15 mW. The optical
filters are of order 7 with bandwidth 100 GHz for MUX/DEMUX. The
transmission results are shown in FIG. 42. It is evident that the
conventional setup suffers a great deal from nonlinearity-induced
amplitude and timing jitters, which are greatly reduced in the system
with scaled translational symmetry, where, however, the ghost-pulse
generation imposes a serious limitation. With both scaled translational
symmetry and mid-span SPM, the third system enjoys a superb signal
quality at the end, with small signal fluctuations due to EDFA noise and
possibly a little residual nonlinearity.

[0125] At the end of the previous section, we have seen that even an
optimized system using SMF+RDF and RDF+SMF spans with "one-for-many"
scaled translational symmetry suffers a great deal of noise in the
originally OFF time slots. A good part of the noise energy there may be
due to the growth of ghost-pulses, which is not suppressed by the
"one-for-many" scaled translational symmetry alone. Naturally, the
above-described method of mid-span SPM may be applied to an optimized
system with "one-for-many" scaled trans-lational symmetry. When
channelized SPM is introduced at the mid-span of the optimized system
depicted in FIG. 37, the resulted setup is shown in FIG. 43. The optical
eye diagram received at the end of the 1280 km transmissions is shown in
FIG. 44, where the ghost-pulses are substantially removed, in sharp
contrast to the top diagram of FIG. 40.

[0126] It should be noted that the present method of ghost-pulse
suppression by mid-span SPM is not limited to transmission lines with
scaled translational symmetries. One or both sides, before or/and after
the channelized SPM, may be configured in mirror symmetry as well for
intra-channel nonlinearity compensation [46, 52], and ghost-pulse
suppression would be just as effective, provided that the two sides
generate nearly the same ghost amplitudes to originally empty data slots.
Moreover, one or both sides may be a general transmission line that is
not optimally designed for intra-channel nonlinearity compensation. In
which case, ghost-pulse generations may still be well suppressed due to
the cancellation of ghost amplitudes generated by the two sides, however
the mark pulses in the originally ON data slots may suffer significant
jitters in amplitude and timing, as a result of the transmission system
being lacking in a (scaled) translational or mirror symmetry.

[0127] It is interesting to compare the present method of mid-span SPM and
signal reshaping based on nonlinear optical loop mirrors (NOLMs) [63,
64], both of which are able to suppress ghost-pulses, and both are
channelized solutions suitable for systems with a high modulation speed,
because where the number of wavelength channels is less and higher
optical power is available in each channel for efficient nonlinear
effects. While a NOLM is often regarded as a lumped signal regenerator,
mid-span SPM may be viewed as a method of distributive signal
regeneration, whose action takes place through an entire transmission
line. Practically, mid-span SPM would be more convenient than NOLMs, as
the latter require interferometry stability and are sensitive to
variations of fiber birefringence [65]. On the other hand, NOLMs are
capable of "removing" random optical noise due to amplified spontaneous
emission and loss-induced quantum noise [66], while mid-span SPM is not.

[0128] Dispersion compensating fibers have become essential components in
high-speed long-distance fiber-optic transmissions. Often they are
packaged into compact DCMs and integrated with fiber optical amplifiers
at the repeater sites. The loss of signal power in DCFs requires extra
gain from optical amplifiers, and amplifiers introduce noise. Because of
their small modal area, DCFs could be significant contributors of
nonlinearity if the power of signals carried is not limited to a low
level. In the past, DCF manufacturers have strived to reduce the loss of
DCFs and to lower their nonlinearity by enlarging the modal area [67].
However, reduced DCF nonlinearity does not necessarily translate into
improved overall transmission performance. In the above, we have
demonstrated that the nonlinear response of DCFs may be taken
advantageously to compensate the nonlinearity of transmission fibers
(TFs). Simply minimizing the loss in such nonlinearity-compensating DCFs
is not necessarily aligned with the best system performance either. Here
we propose and analyze a method of packaging DCFs to achieve optimal
nonlinearity compensation and good signal-to-noise ratio (SNR)
simultaneously. Simply stated, an optimally packaged DCM may consist of
two (or more) portions of DCFs with higher and lower loss coefficients.
In the first portion that experiences high signal power, the loss
coefficient may be intentionally increased in proportion to the DCF
dispersion with respect to a TF. In another portion where the signal
power is low and nonlinearity is negligible, the loss coefficient may be
minimized to output stronger signals while compensating the remaining
dispersion due to the TF.

[0129] Effective nonlinearity compensation between DCFs and TFs, with or
without optical phase conjugation (OPC), relies on careful arrangements
of different types of fibers in a transmission line to form the so-called
scaled translational symmetry. The above has established the analytical
theory and numerical simulations verifying nonlinearity compensation
using translational symmetry. Basically, for two fibers to be matched for
a translational symmetry in the scaled sense about an optical phase
conjugator, their parameters need to obey the following scaling rules,

[α',β2',β3',γ'P0',g'(t)P0']=R[-
α,-β2,β3,γP0,g(t)P0], (49)

where α, β2, β3, γ, and g(t) are the
loss, second-order dispersion, third-order dispersion, Kerr and Raman
nonlinear coefficients respectively for one fiber, while the "primed"
parameters are the corresponding parameters of the other fiber, P0
and P0' are the signal powers input to the two fibers respectively,
R>0 is a scaling factor. Such scaled translational symmetry proves to
enable nonlinearity compensation between the two matched fibers up to the
first-order nonlinear perturbation. The seemingly limited compensation
capability based on perturbation is in fact quite relevant and powerful
in practice, because the nonlinear response of each fiber segment is
indeed perturbative in long-distance transmission lines, and matched
fiber pairs may be arranged in a mirror-symmetric order to effectively
undo the nonlinear distortions that may have accumulated far beyond the
regime of perturbations. In the absence of OPC, a DCF and a TF may still
be arranged into a translational symmetry in the scaled sense according
to the following rules,

(α',β2',β3,γ'P0')=R(α,-β-
2,-β3,γP0), (50)

where again (α, β2, β3, β) and (α',
β2', β3', γ') are parameters of the two types
of fibers respectively. In both cases of scaling rules of equations (49)
and (50), the requirements for the third-order dispersions may be
relaxed, then the two fibers are not in strict translational symmetry
across a band of wavelength channels, rather the symmetry and
nonlinearity compensation between them become approximate. Nevertheless,
such approximation is often a good one when the value of
|β2/β3| is high, so that the percentage change of
β2 is only small across the band, which is exactly the case for
SMFs in the 1550-nm band.

[0130] In our methods of compensating fiber nonlinearity using
translational symmetry with or without optical phase conjugation,
dispersion-compensating fibers are brought into scaled translational
symmetry with respect to TFs such as SMFs and NZDSFs. As noted before, in
regions of dispersion-compensating fibers carrying lower optical power,
the scaling rules of fiber parameters in equations (49) or (50) may be
relaxed without sacrificing the performance of nonlinearity compensation,
both for systems using cabled DCFs into the transmission lines and for
systems using lumped DCMs at the repeater sites. Such relaxation may be
done for practical convenience, or to control the accumulated dispersion
in a span to a desired value, as well as to reduce the span loss so to
reduce the penalty due to optical noise. As illustrated in FIG. 45, a
compact dispersion-compensating module or a dispersion-compensating
transmission line may consist of two portions of dispersion-compensating
fiber concatenated, where the first portion carrying high-power signals
may have an intentionally increased loss coefficient to form a scaled
translational symmetry with a TF, while the second portion experiencing
low signal power could have the minimal loss coefficient and does not
need to satisfy any scaling rule. The two portions of DCF with higher and
minimal loss coefficients may be of one whole piece of fiber coiled with
different radiuses, or differently fabricated DCFs with different loss
coefficients and possibly different dispersions, so long as the first
fiber is in scaled translational symmetry to a target TF. The minimal
loss coefficient refers to the lowest fiber loss coefficient that is
achievable in practical fabrication of dispersion-compensating fibers.
The loss coefficient of the fiber portion on the left side of FIG. 45 is
higher in the sense that it is intentionally made higher than what is
achievable by practical manufacturing processes.

[0131] The great advantage of nonlinearity compensation using scaled
translational symmetry is that a pair of matched fiber segments are
required to have the same sign for the loss/gain coefficients and
opposite dispersions. Such conditions are naturally satisfied in
conventional fiber transmission systems, where a TF, for example an SMF,
may be paired with a DCF as matched counterparts. However, traditional
transmission lines are usually set up with the same configuration for all
spans, that is, with a TF followed by a DCF. Consequently, the
accumulated dispersion in all spans is single-sided, namely, stays always
positive or always negative. Such may be called an M-type dispersion map,
as shown in FIG. 46, where no two spans could form a scaled translational
symmetry. Our proposal is to simply exchange the ordering of the TF and
DCF for some spans, which may be paired with traditional spans to form an
N-type dispersion map, where the accumulated dispersion may go both
positive and negative and trace an N-like curve, as shown in FIG. 47. A
scaled translational symmetry is formed between two matched fiber spans
as in FIG. 47, in the sense that the TF of the first span is scaled
translationally symmetric to the DCF of the second span, and the DCF of
the first span is scaled translationally symmetric to the TF of the
second span. Such translational symmetry between two matched spans could
cancel some of their intra-channel nonlinearities, or compensate all
nonlinearities up to the first-order perturbation if an optical phase
conjugator is installed in the middle. Furthermore, many pairs of matched
spans may be arranged into a mirror-symmetric order about the point of
OPC to form a long-distance transmission line, whose second part could
undo the nonlinear distortions due to the first part that may have
accumulated far beyond the regime of perturbations.

[0132] In traditional transmission lines, each fiber span has a TF and a
DCM at the end, which consists of a conventional DCF with a multi-stage
EDFA. Many such conventional fiber spans are cascaded to form a line with
the M-type dispersion map, as shown on the top of FIG. 48, where a
conventional DCM is denoted by CDCM_M in short. If the order of TF and
DCF is switched for every other span, then an N-type dispersion map is
formed, and two adjacent DCFs may be packaged into one DCM, denoted by
CDCM_N, as shown in the middle of FIG. 48. As a result of the N-type
dispersion map, intra-channel nonlinearities may be suppressed to some
extend, and all fiber nonlinearities may be partially compensated at the
presence of OPC in the middle of the transmission line. However, the
compensation of nonlinearity is not optimal as the scaling rules of
equation (49) or (50) are not satisfied by conventional DCFs paired with
TFs. Indeed, DCF manufacturers have succeeded in reducing the loss of
DCFs, as it was thought to monotonically improve the performance of
transmission systems [67]. The dispersion-to-loss ratio (DLR) of
state-of-the-art DCFs is often much larger than that of TFs. From the
stand point of matched nonlinearity compensation, it would be
advantageous to (intentionally) raise the loss of DCFs such that the DLRs
of DCFs and TFs are matched to satisfy the scaling rules, at least for
portions of fibers carrying high-power signals. On the other hand, in
regions of DCFs experiencing low signal power, the nonlinearity is weak
and negligible, then the scaling rules may be disregarded and the loss of
DCFs may be minimized to enhance the optical SNR at the end of dispersion
compensation. Therefore, an optimized DCM (ODCM), as shown at the bottom
of FIG. 48, may consist of sections of DCFs with higher and lower loss
coefficients, as well as multiple EDFA stages to repeat the signal power
and regulate the signal power in the lossier portions of DCFs, according
to a set of scaling rules with respect to the TFs. Higher DCF loss may be
induced by impurity-doping during fiber manufacturing [32, 33] or by
bending loss during fiber packaging [24].

[0133] Therefore, a DCM compensating the dispersion and nonlinearity of
transmission fibers may be so packaged that the first portion of DCF
experiencing a high level of signal power may have a higher loss
coefficient satisfying the scaling rule in equation (49) or (50), whereas
the second portion of DCF may ignore the scaling rules and become less
lossy such that the signal power at the end of the DCM is not too low to
be significantly impaired by the amplifier noise. In fact, the low-loss
portion of the DCM may even use optical filters other than DCFs, such as
fiber Bragg gratings and photonic integrated circuits. This method of
packaging DCMs achieves the capability of nonlinearity compensation and
good noise performance simultaneously. For instance, it takes 10 km DCF
with D'=-80 ps/nm/km to compensate 100 km NZDSF with dispersion D=8
ps/nm/km and loss α=0.2 dB/km. The first 4 km of the DCF may be
made highly lossy by a special treatment in manufacturing or packaging,
with a loss coefficient α'=2 dB/km to form a scaled translational
symmetry with respect to the first 40 km NZDSF for optimal nonlinearity
compensation. However, the remaining 6 km DCF may ignore the scaling
rules and have a much lower nominal loss α'=0.6 dB/km [4]. The
total loss is reduced by 8.4 dB as compared to a DCM that complies
strictly with the scaling rules throughout the length of the DCF. Another
important parameter of DCFs is the effective modal area, or more directly
the nonlinear coefficient. Traditional designs of DCFs have always
strived to enlarge the modal area so to reduce the nonlinear effects of
DCFs. However, for DCFs used in our methods of nonlinearity compensation,
there exists an optimal range of modal area which should be neither too
large nor too small. According to the scaling rules of equation (49) or
(50), a DCF with a large modal area may require too much signal power to
generate sufficient nonlinearity to compensate the nonlinear effects of a
transmission fiber, while optical amplifiers may have difficulty to
produce that much signal power. On the other hand, when the effective
modal area is too small, the scaling rules of equation (49) or (50)
dictate a reduced power level for the optical signal in the DCF, which
may be more seriously degraded by optical noise, such as loss-induced
quantum noise [66] and the amplified-spontaneous-emission noise from an
amplifier at the end of the DCF.

[0134] To give an example of ODCM and test its performance in nonlinearity
compensation, we simulated (using VPItransmissionMaker®) and compared
three transmission systems as shown in FIGS. 49, 50, and 51 respectively,
all of which have an optical phase conjugator in the middle and 6
recirculating loops on each side of OPC. For the first system, each
recirculating loop consists of two identical spans of 100 km SMF followed
by a CDCM_M, as shown on the top of FIG. 48. For the second system, each
recirculating loop has 100 km SMF followed by a CDCM_N, then 100 km SMF
followed by a 20 dB EDFA, as shown in the middle of FIG. 48. For the
third and optimized system, each recirculating loop consists of 100 km
SMF followed by an ODCM, then 100 km SMF followed by a 20 dB EDFA, as
shown at the bottom of FIG. 48. Each CDCM_M has a 15 dB EDFA, 20 km
conventional DCF, then a 15 dB EDFA. Each CDCM_N has a 15 dB EDFA, 20 km
conventional DCF, then another 15 dB EDFA, 20 km conventional DCF, and
finally a 10 dB EDFA. By contrast, each ODCM consists of a 21 dB EDFA, 10
km optimized DCF, 10 km conventional DCF, a 14 dB EDFA, then 10 km
optimized DCF, 10 km conventional DCF, and a 15 dB EDFA. Note the
adjustment of signal power in the optimized DCFs to fulfil the scaling
rules. The SMF has loss α=0.2 dB/km, dispersion D=16 ps/nm/km,
dispersion slope S=0.055 ps/nm2/km, effective modal area
Aeff=80 μm2. The conventional DCF has (a', D', S',
Aeff')=(0.5, -80, -0.275, 20) in the same units. The optimized DCF
differs from the conventional one only by the loss coefficient
α''=1.0 dB/km. The same silica nonlinear index
n2=2.6×10-20 m2/W is taken for all fibers. All EDFAs
have the same noise figure of 4 dB. The center frequency is 193.1 THz.
The inputs are four 40 Gb/s channels, spaced by 200 GHz, co-polarized and
return-to-zero modulated with 33% duty and pulse peak power 15 mW. The
eye diagrams of optical signals at the end of transmissions are shown in
FIG. 52, where the top diagram displays severe nonlinear distortions for
the conventional line with the M-type dispersion map, while the middle
diagram shows improved but still seriously impaired signals of the line
with the N-type dispersion map using conventional DCMs. The bottom
diagram demonstrates a significant improvement of signal quality by using
optimized DCMs and scaled translational symmetry, where the signal
distortions are mainly due to EDFA noise and possibly some uncompensated
nonlinearity.

[0135] Even without OPC, improved transmission performance due to
intra-channel nonlinearity compensation may be expected in transmission
systems manifesting scaled translational symmetries using optimally
packaged DCMs for matched nonlinear compensation and reduced optical
noise simultaneously. Furthermore, the method of mid-span SPM discussed
previously may be employed in such transmission systems using ODCMs to
suppress the generation of ghost-pulses, which are not controlled by
scaled translational symmetries alone. Finally, it is noted that the same
principle for optimally packaging DCMs, namely, obeying scaling rules
where the signal power is high while disregarding the rules and
minimizing the signal loss where the signal power is low, may be
similarly applied to the design of transmission systems with cabled DCFs.
For a piece of DCF cabled into a transmission line, the first portion of
the DCF may have a relatively low absolute value of dispersion in
proportion to its low loss coefficient, according to the scaling rules of
translational symmetry to a trans-mission fiber as in equation (49) or
(50). Whereas in the second portion of the DCF, where the signal power
becomes sufficiently low, the dispersion may be set as high as possible
while the loss coefficient should remain minimal, because no scaling
rules need to be regarded.

APPENDIX

Fiber Parameters

[0136] Using the D and S parameters carelessly can lead to confusion. For
instance, the values D=16 ps/nm/km and S=0.08 ps/nm2/km are often
cited for the standard single-mode fiber. We note that it is necessary to
use the D and S values at the same wavelength for the same fiber to avoid
confusion. At 1550 nm, the SMF has D≈16 ps/nm/km and
S≈0.055 ps/nm2/km instead of 0.08 ps/nm2/km, which is
the approximate dispersion slope at 1310 nm. Regarding the use of D and S
in simulations, our scaling rules are for β2 and β3,
not directly D and S. The relations are given by,

By contrast, for perfect direct (without OPC) dispersion compensation,
the compensating fiber should have parameters -(β2,
β3), and correspondingly -(D, S). When the scaling factor is
not one, the parameters of the compensating fibers should multiply
whatever the ratio R>0, for all the three cases.

[0138] Another important parameter is the effective modal area Aeff,
often specified alternatively by the mode field diameter (MFD). The MFD
is defined as the diameter of the circle where the optical intensity
decays to 1/e of the peak value. If the modal field is approximated as
Gaussian, then there is the relation,